We determine explicit variational expressions for the quantum free energy of mean-field
spin glasses in a transversal magnetic field, whose glass interaction is given by a
hierarchical Gaussian potential as in Derrida’s generalized random energy model
(GREM), its continuous version (CREM), or the nonhierarchical GREM. The
corresponding phase diagrams, which generally include glass transitions as well as
transitions to quantum paramagnetic and mixed quantum-classical phases, are
discussed. In the glass phase, the free energy is generally determined by both the
parameters of the classical model and the transversal field.