Actually, the Sees of Antioch and Alexandria are far older and more prestigious than the sees of Constantinople and even Jerusalem, during the opening stages of Christianity as the state religion of the Roman Empire, at least.

Rome, Alexandria and Antioch were actually the original three Sees, or the Petrine sees, because according to the text of the First Council of Nicaea, these three places are the seats of the Metropolitans of the Christian church, with Rome presiding over the entirety of Western Europe, Antioch presiding over the "East" (which probably means modern day Syria, Iraq and Western Turkey and the Caucasus) and Alexandria leading the faithful in the "Diocese in Egypt" which is modern-day Egypt, Libya and a bit of Saudi Arabia. The reasons why are:

• Rome was where Peter was martyred by Nero, and thus, if the theorists of Papal Primacy are to be believed, cemented Rome as the place where the rock of the church is by virtue of having Peter being buried beneath it, and thus was (according to the defenders of Papal Primacy) the most important See.

• Antioch was where the Incident of Antioch took place, (as described in Paul's Epistle to the Galatians 2:11-14) where Peter had a theological debate on how the Gentiles were being converted to Christianity, specifically the debate whether the gentiles really had to conform to the traditional Jewish/Abrahamic institution of circumcision in order to be truly "Christian". This argument made Antioch a more important See than Alexandria but less so than Rome.

• Alexandria was where Peter's student, Mark the Evangelist, well, evangelized, and thus the least important among the three.

Jerusalem at the creation of the First Council of Chalcedon was largely a ruin, just having been razed to the ground during the Jewish Revolts and renamed to Aealia Capitolina, and so was only given honorary rights to a see in the First Council of Nicaea, but finally granted full rights to a See when Justinian I instituted the Pentarchy.

Constantinople didn't have a strong theological background on being an actual See, but seeing that it was New Rome, and thus the capital of the Empire after Constantine I moved it from Rome, it had the backing of the Emperor and was thus elevated to an important See in the Council of Chalcedon.

Source: Schaff, P. (Ed.). (n.d.). NPNF2-14. The Seven Ecumenical Councils. Retrieved December 21, 2017, from http://www.ccel.org/ccel/schaff/npnf214.toc.html#P3825_663549

P.S. This might be a question better answered in /r/Catholicism