  Lyapunov exponents, a purely classical quantity, play an important role in
the evolution of quantum chaotic systems in the semiclassical limit. We
conjecture the existence of an upper bound on the Lyapunov exponents that
contribute to the quantum motion, namely, even in the semiclassical limit only
a limited range of Lyapunov exponents, bounded from above, are important for
the quantum evolution. This is a universal feature in any quantum system or
quantum field theory, including those with a gravity dual. It has its origin in
the finite size of the Hilbert space that is available to an initial
quasi-classical configuration. An upper bound also exists in the limit of an
infinite Hilbert space provided that the system is in contact with an
environment, for instance a thermal bath. An important consequence of this
result is a universal quantum bound on the maximum growth rate of the
entanglement entropy at zero and finite temperature.
