Within last years multi-synchronous collaborative editing systems became widely used. Multi-synchronous collaboration maintains multiple, simultaneous streams of activity which continually diverge and synchronized. These streams of activity are represented by means of logs of operations, i.e. user modifications. A malicious user might tamper his log of operations. At the moment of synchronization with other streams, the tampered log might generate wrong results. In this paper, we propose a solution relying on hash-chain based authenticators for authenticating logs that ensure the authenticity, the integrity of logs, and the user accountability. We present algorithms to construct authenticators and verify logs. We prove their correctness and provide theoretical and practical evaluations.