When Alexander departed for the east in 334 BCE, he took a gigantic portion of "Macedon" with him: the male soldiers of the armies represented the political power of the Macedonian state, not to mention the vast retinue of aristocrats who also went along. Upon the death of Darius, Alexander had to convince his army to continue, and he had to deal with them as if they were the Macedonian state. But he had left behind Antipater to look after the homeland and to be strategos (general) of Europe in his absence. Antipater had been a member of Philip II's old guard, and was a very capable fellow.
Alexander's destruction of Thebes had left Greece in a very weakened and peaceful state upon the departure to the East. The vast amounts of booty returning to Europe from Alexander's battles keep the Greek cities prosperous and quiet, for the most part. There were initial worries about the powerful Persian fleet left behind in the Aegean, but that threat evaporated after Issus in 333, less than a year into the campaign. Its removal rid Antipater of a major logistical headache in the making. Meanwhile, the Thracian tribes, led by their Macedonian overseer Memnon, decided to rebel. They were handled without much apparent fuss and without comment in 332. This was probably the third-most-serious threat to Antipater's governing. A year later, in 331, the Spartans under king Agis revolted and initially created quite a big stir. Improbably, however, Antipater defeated them at Megalopolis with a hodgepodge force of mercenaries and Macedonian reserves. He even referred the matter to Philip's League of Corinth instead of acting as an overbearing autocrat. Between the Persian fleet, the Thracian revolt, and then the Spartan revot, he really managed a series of crises well and should be remembered for his efforts. Alexander seems to have been jealous of Antipater's handling of the Spartan revolt.
The biggest threat to Macedonian political equanimity came from Olympias, Alexander's powerful and influential mother. She and Antipater had been allies in earlier times, but now became bitterly opposed to one another. Olympias came to wield a significant amount of influence, both through her proxy as Alexander's mother, and in her own right. Alexander had to deal with them both, and what seems to have been their frequent squabbles back home. Through all this, the lines of communication between Pella and the East remained open and reliable, so information and orders passed back and forth freely, if slowly. But Antipater had things thoroughly handled back home, giving Alexander the relative freedom to do what he wanted (which he always did anyway).