  We introduce a simple quantum generalization of the spectrum of classical
Lyapunov exponents. We apply it to the SYK and XXZ models, and study the
Lyapunov growth and entropy production. Our numerical results suggest that a
black hole is not just the fastest scrambler, but also the fastest entropy
generator. We also study the statistical features of the quantum Lyapunov
spectrum and find universal random matrix behavior, which resembles the
recently-found universality in classical chaos. The random matrix behavior is
lost when the system is deformed away from chaos, towards integrability or a
many-body localized phase. We propose that quantum systems holographically dual
to gravity satisfy this universality in a strong form. We further argue that
the quantum Lyapunov spectrum contains important additional information beyond
the largest Lyapunov exponent and hence provides us with a better
characterization of chaos in quantum systems.
