  We study first order quantum phase transitions in mean-field spin glasses. We
solve the quantum Random Energy Model using elementary methods and show that at
the transition the eigenstate suddenly projects onto the unperturbed ground
state and that the gap between the lowest states is exponentially small in the
system size. We argue that this is a generic feature of all `Random First
Order' models, which includes benchmarks such as random satisfiability. We
introduce a two-time instanton to calculate this gap in general, and discuss
the consequences for quantum annealing.
