  In this paper we study the out-of-equilibrium phase diagram of the quantum
version of Derrida's Random Energy Model, which is the simplest model of
mean-field spin glasses. We interpret its corresponding quantum dynamics in
Fock space as a one-particle problem in very high dimension to which we apply
different theoretical methods tailored for high-dimensional lattices: the
Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and
the cavity method. Our results indicate the existence of two transition lines
and three distinct dynamical phases: a completely many-body localized phase at
low energy, a fully ergodic phase at high energy, and a multifractal "bad
metal" phase at intermediate energy. In the latter, eigenfunctions occupy a
diverging volume, yet an exponentially vanishing fraction of the total Hilbert
space. We discuss the limitations of our approximations and the relationship
with previous studies.
