  Periodically-driven systems are ubiquitous in science and technology. In
quantum dynamics, even a small number of periodically-driven spins leads to
complicated dynamics. Hence, it is of interest to understand what constraints
such dynamics must satisfy. We derive a set of constraints for each number of
cycles. For pure initial states, the observable being constrained is the
recurrence probability. We use our constraints for detecting undesired coupling
to unaccounted environments and drifts in the driving parameters. To illustrate
the relevance of these results for modern quantum systems we demonstrate our
findings experimentally on a trapped-ion quantum computer, and on various IBM
quantum computers. Specifically, we provide two experimental examples where
these constraints surpass fundamental bounds associated with known one-cycle
constraints. This scheme can potentially be used to detect the effect of the
environment in quantum circuits that cannot be classically simulated. Finally,
we show that, in practice, testing an $n$-cycle constraint requires executing
only $O(\sqrt{n})$ cycles, which makes the evaluation of constraints associated
with hundreds of cycles realistic.
