This is gonna be a long answer, be warned. Tsarist Russia had a long history of anti-semitism that often resulted in violent pogroms. When the Bolsheviks seized power, the new USSR was formed. Lenin staunchly criticised anti-semitism. Trotsky, Kamenev, and Kaganovich were all major Jewish Bolsheviks.

After Lenin died however, Stalin seized power. He attacked the "Trotskyite" factions and nurtured a strong hatred of Trotsky. Trotsky was forced into exile in 1929. Stalin adopted increasingly harsh measures towards Jews as the years went on. Jews serving in the Soviet NKVD (Jews were overrepresented in it) suffered massively during the Great Purges. From 1937 to July 1939, Jews went from being 38% of the NKVD to just 4%. Russians grew to 67% and from then on were predominant.

During and after the War, the Holocaust of Jews was downplayed by Stalin's government and instead the Soviets as a whole were seen as the victims. Soviet accounts of the Babi Yar massacre edited out references to its anti-semitic character. The Jewish Anti-Fascist Committee, which tried to point out the specific mass murder of Jews, was prosecuted and many of its members killed. A post-war campaign against "Zionists and rootless cosmopolitanism" resulted in many Jewish figures (including poets and academics) being arrested or executed. This coincided with the USSR moving away from Israel to back Arab regimes.

Now the question people are divided over is whether Stalin's actions were motivated by anti-semitism. Some believe they were. They see the feud with Trotsky as setting off Stalin's anti-semitism. Stalin was also known to mock Jews in private . But others believe Stalin was mostly advancing his own political goals and didn't harbor any especially strong anti-semitism. He stayed close to Kaganovich. He mocked and persecuted other minorities as well. He originally supported the creation of Israel.

Personally I see Stalin as being primarily driven by a need for political control, not racial hatred. Evidence: even as he exacerbated the Holodomor famine that killed 3.4 million Ukrainians, he also oversaw Ukrainian membership in the Party and the NKVD rise significantly.

Sources:

Bloodlands by Timothy Snyder

Berlin by Antony Beevor

Stalin: Court of the Red Tsar by Sebag Montefiore

Edited for clarity

Note, this is taken from an earlier answer here and here

The tsarist state considered the Jews that resided within its empire as subjects to the crown, and increasingly fell under the category of inorodtsy in the nineteenth century (inorodtsy essentially means subject aliens). The majority of the empire's Jews lived in a region called the Pale of Settlement, although there were Jewish communities within the Baltic and Russian areas. Most Jews could not migrate outside of the Pale unless they had a profession or other factor in their favor such as military service (unlike other inordtsy groups, Jews were subject to conscription in the nineteenth century). Thus the Jews outside of the Pale (Lithuanian Jews being a notable exception- it's complicated) tended to acculturate to Russian mores. As the nineteenth century wore on, political activism among Jews within the Pale tended to drift towards either Zionism or Bundism. The former was a form of Jewish nationalism that sought to revive Hebrew and held Jewish culture as distinct from neighboring ethnic groups. Bundists also hewed to Jewish particularism, but within a Marxist-socialist context and they found a receptive audience in the Pale's major cities like Lodz, which were rapidly industrializing and had a strong Jewish population. Although traditionalist Jews within the Pale were a strong element of society, both Zionism and the Bundists received a boost in membership from the pogroms of the late nineteenth century. This anti-Jewish violence fostered counter-organization as the tsarist state was unwilling or unable to protect the lives of its Jewish subjects. The First World War and the dislocation of the Civil War led to even more pogroms and antisemitism flourished among White forces.

Although Zionists made some tactical alliances with revolutionary forces in the changed situation of 1917, the Bundists found it much easier to ally with the Reds. A number of Jews also flocked towards the Bolsheviks for a number of reasons. The Red Army in many ways put a stop to the pogroms within Ukraine and the Bolsheviks eliminated the tsarist legal restrictions on Jews' movement and occupation. As the Soviet Union emerged from civil war, it found it needed a trained and educated bureaucratic civil service and many Jews, often Bundists, staffed these lower- and mid-level positions since Jews by and large had higher literacy rates than other members of the population. The Soviet state also encouraged Jewish cultural projects and held that all its citizens were equal before the law irrespective of ethnicity or religion.

It is tempting to view the 1920s through an idealized lens for Soviet-Jewish relations, however such idealizations are deceptive. The crux of the "Jewish Question" was over whether or not being Jewish was an ethnicity or form a religious practice. If one ascribed to the idea that Jewish identity was ethnicity, then Jews should be welcomed into the brotherhood of Soviet nations; conversely, if Judaism was a religious practice, then it was an archaic form of identity that could never square with the new Soviet man the USSR was creating. The fractious nature of this debate was apparent in the special Jewish section of the Communist Party (evsektsii) which unlike other special ethnic sections was explicitly temporary. By the late 1920s, the Stalin turn saw a partial victory of the perspective that Jewish was a religious practice and many of these special privileges (Terry Martin terms this Soviet-style Affirmative Action) granted to Jews were repealed. Stalin also arguably appealed to a coded antisemitism in deposing of his rivals like Trotsky, who had a taint of a Jewish background.

Although Soviet Jews found the 1920s privileges curtailed under Stalin, the state did not openly hew towards an antisemitic line. Open statements of antisemitism were disciplinary offenses if reported to the state's authorities. The rise of the Third Reich also precluded a open antisemitism as German racism became lumped together as an ancillary component of Nazism's anticommunism within Soviet propaganda. Majority Jewish areas like Minsk openly embraced their identity as Soviet Jews as the ideological war between communism and fascism sharpened during the late 1930s. Despite this early form of Soviet patriotism, many Jews became victims of the Terror.

The Great Purges of the late 1930s had a large number of Jewish victims. Some of this was due to a covert antisemitism amongst security officials, but keep in mind many of the purgers were themselves Jewish in origin. Part of the reason for the number of Jewish victims is a function of how the Purges were conducted. If one had a questionable ideological background pre-1917, the odds of surviving the Purges lessened a great deal. Since many Jews involved in the Soviet state were former Bundists, they had a black mark on their biographical commitment to revolution that they could not erase like their gentile colleagues. Furthermore, informal antisemitism also meant that many non-Jews within the state saw Jews as outsiders, which was dangerous in the context of the circular firing squad of the Purges. Many of the mid-level managers when asked to name names by security forces would sacrifice outsiders before they would give up their inner circle.

The Second World War also saw an uptick in Russian antisemitism as the state no longer bothered to discipline antisemitic outbursts among the public. The life or death struggle with the Third Reich also precluded any acknowledgement by the USSR that antisemitism existed within its own borders because that would mean the Soviet Union shared common ground with its existential foe. A rekindling of antisemitism also received further sanction from the upper echelons of Soviet society as Stalin's paranoia directed itself towards Jewish elements within the USSR, culminating in the famous Doctor's Plot. Despite this postwar upswing, antisemitism within the USSR shared little with the antisemitism of Nazi Germany; Jews were still Soviet citizens and if they acculturated, they faced little formal discrimination by the state. That being said, Jews were an ethnicity that the postwar Soviet state possessed little love for, and the state was indifferent to complaints about antisemitism among the general Soviet public. The Cold War and the Arab-Israeli conflict added a further strain as the Soviets increasingly backed the Arabs in their public diplomacy. This meant that Zionism became linked in Soviet discourse with Western imperialism, so then to emphasize being a Soviet Jew during the postwar meant identifying with an ideological opponent of the state, unlike the 1930s when being a Soviet Jew was doubly antithetical to Nazi Germany.

Soviet discourses on the Holocaust demonstrate the outsider status of Jew. As a historical topic, the murder of Jews in the USSR was not so much forbidden as it was downplayed and minimized. The Soviet approach to war commemoration throughout the postwar era emphasized that Soviet Communism was both the primary victim and target of fascism. Thus massacre sites like Babi Yar, a ravine in which upwards of a 100000 Jews were shot, were not unknown in the Soviet Union, but were instead commemorated as areas in which the Germans massacred thousands of Soviet civilians. This is of course functionally true, the Jewish victims of the Einsatzgruppen were indeed citizens of the USSR, but this generalization enfolded the Nazi’s Jewish massacres into a wider narrative of Soviet victimization and eventual triumph. The initial draft of the discovery of Babi Yar for the Soviet press indicated that a majority of its victims was Jewish, but the official memorandum deleted these specifics.

The rationale for this minimization goes beyond just the Soviet desire to be the primary victim of Nazism. The Soviet ideological conceptualization of antisemitism was that it was a superstructure of the wider antipathies towards Marxism within the capitalist West. In Soviet orthodoxy, antisemitism was a tool used by the enemies of Marxism, along with religion, racism, or nationalism, to prevent support for Marxist-Leninist movements among various disaffected groups. The Nazi apparatus for extermination of Jews was something this Soviet understanding of antisemitism was intellectually unprepared for. The Soviet state was also very ambivalent towards Judaism in general. Consequently, asserting that Jews were a victim independent of the wider actions of the Third Reich stood against the grain of the Soviet stance towards Judaism.

This is not to say that some cracks in this state-centric memory did not appear during the postwar period. One Soviet poet, Yevgeny Yevtushchenko wrote “Babi Yar” in 1961 that connected both the Jewish nature of the massacre with the wider policies of extermination such as the death of Anne Frank. Other cultural depictions of the war recognized Jewish suffering, but were much more low-key in that acknowledgement. For example, the 1965 film Ordinary Fascism has a harrowing scene in Auschwitz, but does not specifically identify its victims as Jews and asserts that had the Germans been successful, Himmler would have expanded the camps into the Soviet Union. But it would take Glasnost and the collapse of the Soviet Union to undermine this edifice that prioritized Soviet suffering over all others.

Are you referring to certain actions taken during the days of the USSR because in my understanding Russia has had a long history of antisemitism. The Pogroms of the early 1800s against Jews (which continued into the 20th century by groups like the Black Hundreds). Also Jews were only allowed to live and do business in a region known as the Pale of Settlement. Also, in regards to American/Russian relations, ethnicity was required on passports in Russia and caused a rift in state relations at the onset of WWI as Russian Jews who emigrated to America that went back to Russia on American passports were dealing with discrimination because they were Jewish. This topic is gone more in depth in The First Cold War: The Legacy of Woodrow Wilson in U. S. -Soviet Relations By Donald E. Davis, Eugene P. Trani

It seems to me that imperial russia and its successor state the ussr only stopped prosecuting jewish people for a while after the revolution itself and then went somewhat back to 'normal' with stalin who had his doctors jailed for being jewish (even though they were technically atheists, just like him).

The tsars were only occasionally fair with the jewish population, catherine the great basically turned all polish jews into second or even third-rate citizens with stricts limits to where they could live and even which trades they could learn and practice. It took nearly 200 years for tsar alexander II to show up and remove those limitations. At the time you couldn't even find a carpenter in those communities since if you were jewish learning carpentry was forbidden.

Immediately after the soviet revolution restrictions were lifted not only for jews but for many other minorities and groups. However stalin was a nationalist and basically rolled back a lot of these reforms. For example while after the revolution there was a certain tolerance for openly homosexual people after stalin homosexuality became a disease, allegedly "created" by the bourgeois to undermine the revolution, a view that was shared by practically all other communist regimes in the world.

Still stalin view of jewish people became the standard and paved the way for future events to come like the ussr backing the arabs rather than israelis even though the practice of kibbutz and other policies in israel were far closer to socialism than capitalism, unlike many arab states which were also staunchly anti-communists.