  Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes
for chaos in interacting quantum systems. They exhibit a characteristic
classical exponential growth, but saturate beyond the so-called scrambling or
Ehrenfest time $\tau_{\rm E}$ in the quantum correlated regime. Here we present
a path-integral approach for the entire time evolution of OTOCs for bosonic
$N$-particle systems. We first show how the growth of OTOCs up to $\tau_{\rm E}
= (1/\lambda) \log N$ is related to the Lyapunov exponent $\lambda$ of the
corresponding chaotic mean-field dynamics in the semiclassical large-$N$ limit.
Beyond $\tau_{\rm E}$, where simple mean-field approaches break down, we
identify the underlying quantum mechanism responsible for the saturation. To
this end we express OTOCs by coherent sums over contributions from different
mean-field solutions and compute the dominant many-body interference term
amongst them. Our method further applies to the complementary semiclassical
limit $\hbar \rightarrow 0$ for fixed $N$, including quantum-chaotic single-
and few-particle systems.
