Not exactly. You probably mention the socialism. According Marx and Engels the communism and socialsm were synonymous terms. But Lenin divided them. The socialism was "... according to everyone's work", while communism "... everyone's needs"

The communism idea was the "classless society" and "everyone is equal". (unfortunately we had to learn about this in the school).

But in this case, when everyone is equal, then there are no "leaders". The conequence: without leaders there is no state. The communism is an "ouroborous". It's aim to eliminate itself (which is a crazy concept). So if no state exists remains the anarchy (see George Kennan's book titled "Around the cragged hill").