  Quantum chaotic interacting $N$-particle systems are assumed to show fast and
irreversible spreading of quantum information on short (Ehrenfest) time scales
$\sim\!\log N$. Here we show that, near criticality, certain many-body systems
exhibit fast initial scrambling, followed subsequently by oscillatory behavior
between reentrant localization and delocalization of information in Hilbert
space. We consider both integrable and nonintegrable quantum critical bosonic
systems with attractive contact interaction that exhibit locally unstable
dynamics in the corresponding many-body phase space of the large-$N$ limit.
Semiclassical quantization of the latter accounts for many-body correlations in
excellent agreement with simulations. Most notably, it predicts an
asymptotically constant local level spacing $\hbar/\tau$, again given by
$\tau\! \sim\! \log N$. This unique timescale governs the long-time behavior of
out-of-time-order correlators that feature quasi-periodic recurrences
indicating reversibility.
