A concurrent system of synchronous communicating agents is assembled from simpler sequential agents by parallel composition and hiding. For example, hide a1, ... al in (p1 ∥ p2 ⋯ ∥ pn) describes the system of communicating agents p1, ... pn in which the communication events a1, ... al are hidden. Consider descriptions of two systems p and q of synchronously communicating finite state agents. Assume that one wants to check whether p∼q for one of the commonly used equivalence ∼. We show that this question is PSPACE hard for all equivalences that lie between strong bisimulation and trace equivalences. For some equivalences exponential lower and upper bounds are proven. We also show that this problem is NP hard and co-NP hard even for a class of very simple finite agents.