  Synchronization blockade refers to an interferometric cancellation of quantum
synchronization. In this manuscript, we show how the choice of synchronization
measure and Hamiltonian symmetries affect the discussion of synchronization
blockade. Using counting principles, we prove a general theorem that
synchronization blockade cannot be observed in an $N-$level system when the
coherent state used to define the diagonal limit-cycle state is in the full
$SU(N)$ group. We present several illustrative examples of synchronization
blockade in multi-level systems and prove that information-theoretic measures
of synchronization can also observe synchronization blockade-like behavior by
an appropriate choice of the set of limit cycle states.
