  A fundamental result in modern quantum chaos theory is the
Maldacena-Shenker-Stanford upper bound on the growth of out-of-time-order
correlators, whose infinite-temperature limit is related to the operator-space
entanglement entropy of the evolution operator. Here we show that, for
one-dimensional quantum cellular automata (QCA), there exists a lower bound on
quantum chaos quantified by such entanglement entropy. This lower bound is
equal to twice the index of the QCA, which is a topological invariant that
measures the chirality of information flow, and holds for all the R\'enyi
entropies, with its strongest R\'enyi-$\infty$ version being tight. The
rigorous bound rules out the possibility of any sublinear entanglement growth
behavior, showing in particular that many-body localization is forbidden for
unitary evolutions displaying nonzero index. Since the R\'enyi entropy is
measurable, our findings have direct experimental relevance. Our result is
robust against exponential tails which naturally appear in quantum dynamics
generated by local Hamiltonians.
