  Out-of-time-ordered correlation functions (OTOC's) are presently being
extensively debated as quantifiers of dynamical chaos in interacting quantum
many-body systems. We argue that in quantum spin and fermionic systems, where
all local operators are bounded, an OTOC of local observables is bounded as
well and thus its exponential growth is merely transient. As a better measure
of quantum chaos in such systems, we propose, and study, the density of the
OTOC of extensive sums of local observables, which can exhibit indefinite
growth in the thermodynamic limit. We demonstrate this for the kicked quantum
Ising model by using large-scale numerical results and an analytic solution in
the integrable regime. In a generic case, we observe the growth of the OTOC
density to be linear in time. We prove that this density in general, locally
interacting, non-integrable quantum spin and fermionic dynamical systems
exhibits growth that is at most polynomial in time---a phenomenon, which we
term weak quantum chaos. In the special case of the model being integrable and
the observables under consideration quadratic, the OTOC density saturates to a
plateau.
