  It is widely expected that systems which fully thermalize are chaotic in the
sense of exhibiting random-matrix statistics of their energy level spacings,
whereas integrable systems exhibit Poissonian statistics. In this paper, we
investigate a third class: spin glasses. These systems are partially chaotic
but do not achieve full thermalization due to large free energy barriers. We
examine the level spacing statistics of a canonical infinite-range quantum spin
glass, the quantum $p$-spherical model, using an analytic path integral
approach. We find statistics consistent with a direct sum of independent random
matrices, and show that the number of such matrices is equal to the number of
distinct metastable configurations -- the exponential of the spin glass
"complexity" as obtained from the quantum Thouless-Anderson-Palmer equations.
We also consider the statistical properties of the complexity itself and
identify a set of contributions to the path integral which suggest a Poissonian
distribution for the number of metastable configurations. Our results show that
level spacing statistics can probe the ergodicity-breaking in quantum spin
glasses and provide a way to generalize the notion of spin glass complexity
beyond models with a semi-classical limit.
