  The relaxation of out-of-time-ordered correlators (OTOCs) has been studied as
a mean to characterize the scrambling properties of a quantum system. We show
that the presence of local conserved quantities typically results in, at the
fastest, an algebraic relaxation of the OTOC provided (i) the dynamics is local
and (ii) the system follows the eigenstate thermalization hypothesis. Our
result relies on the algebraic scaling of the infinite-time value of OTOCs with
system size, which is typical in thermalizing systems with local conserved
quantities, and on the existence of finite speed of propagation of correlations
for finite-range-interaction systems. We show that time-independence of the
Hamiltonian is not necessary as the above conditions (i) and (ii) can occur in
time-dependent systems, both periodic or aperiodic. We also remark that our
result can be extended to systems with power-law interactions.
