  It was proposed recently that the out-of-time-ordered four-point correlator
(OTOC) may serve as a useful characteristic of quantum-chaotic behavior,
because in the semi-classical limit, $\hbar \to 0$, its rate of exponential
growth resembles the classical Lyapunov exponent. Here, we calculate the
four-point correlator, $C(t)$, for the classical and quantum kicked rotor -- a
textbook driven chaotic system -- and compare its growth rate at initial times
with the standard definition of the classical Lyapunov exponent. Using both
quantum and classical arguments, we show that the OTOC's growth rate and the
Lyapunov exponent are in general distinct quantities, corresponding to the
logarithm of phase-space averaged divergence rate of classical trajectories and
to the phase-space average of the logarithm, respectively. The difference
appears to be more pronounced in the regime of low kicking strength $K$, where
no classical chaos exists globally. In this case, the Lyapunov exponent quickly
decreases as $K \to 0$, while the OTOC's growth rate may decrease much slower
showing higher sensitivity to small chaotic islands in the phase space. We also
show that the quantum correlator as a function of time exhibits a clear
singularity at the Ehrenfest time $t_E$: transitioning from a time-independent
value of $t^{-1} \ln{C(t)}$ at $t < t_E$ to its monotonous decrease with time
at $t>t_E$. We note that the underlying physics here is the same as in the
theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54,
14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)]
and is due to a delay in the onset of quantum interference effects, which occur
sharply at a time of the order of the Ehrenfest time.
