Abstract
We study conservation laws of a general class of quantum many-body systems subjected to an external time dependent quasi-periodic driving. {When the frequency of the driving is large enough or the strength of the driving is small enough, we prove a Nekhoroshev-type stability result: we show that the system exhibits a prethermal state for stretched exponentially long times in the perturbative parameter}. Moreover, we prove the quasi-conservation of the constants of motion of the unperturbed Hamiltonian and we analyze their physical meaning in examples of relevance to condensed matter and statistical physics.