  We study a large-$N$ bosonic quantum mechanical sigma-model with a spherical
target space subject to disordered interactions, more colloquially known as the
$p$-spin spherical model. Replica symmetry is broken at low temperatures and
for sufficiently weak quantum fluctuations, which drives the system into a spin
glass phase. The first half of this paper is dedicated to a discussion of this
model's thermodynamics, with particular emphasis on the marginally stable spin
glass. This phase exhibits an emergent conformal symmetry in the strong
coupling regime, which dictates its thermodynamic properties. It is associated
with an extensive number of nearby states in the free energy landscape. We
discuss in detail an elegant approximate solution to the spin glass equations,
which interpolates between the conformal regime and an ultraviolet-complete
short distance solution. In the second half of this paper we explore the
real-time dynamics of the model and uncover quantum chaos as measured by
out-of-time-order four-point functions, both numerically and analytically. We
find exponential Lyapunov growth, which intricately depends on the model's
couplings and becomes strongest in the quantum critical regime. We emphasize
that the spin glass phase also exhibits quantum chaos, albeit with
parametrically smaller Lyapunov exponent than in the replica symmetric phase.
An analytical calculation in the marginal spin glass phase suggests that this
Lyapunov exponent vanishes in a particular infinite coupling limit. We comment
on the potential meaning of these observations from the perspective of
holography.
