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 (root n) cycles, which makes the evaluation of constraints associated with hundreds of cycles realistic.