  We discuss the connection between the out-of-time-ordered correlator and the
number of harmonics of the phase-space Wigner distribution function. In
particular, we show that both quantities grow exponentially for chaotic
dynamics, with a rate determined by the largest Lyapunov exponent of the
underlying classical dynamics, and algebraically -- linearly or quadratically
-- for integrable dynamics. It is then possible to use such quantities to
detect in the time domain the integrability to chaos crossover in many-body
quantum systems.

Quantum Physics;Statistical Mechanics