We prove an analogue of the “bottleneck theorem”, well-known for classical Markov chains, for Markovian quantum channels. In particular, we show that if two regions (subspaces) of Hilbert space are separated by a region that has very low weight in the channel’s steady state, then states initialized on one side of this barrier will be stuck for a long time. This puts a lower bound on the mixing time in terms of an appropriately defined “quantum bottleneck ratio”, which involves both diagonal and off-diagonal matrix elements of the steady state density matrix. Specializing to the case of finite temperature Gibbs states of a quantum many-body systems, the bottleneck theorem provides a novel, dynamical perspective on non-trivial phases of matter in terms of a decomposition of the Gibbs state into multiple components separated by bottlenecks. We use this perspective to motivate the definition of "topological quantum spin glass" (TQSG) order, which combines features of certain classical spin glass models with those of topologically ordered systems. We prove that TQSG order is realized at low temperatures in a large class of quantum low-density parity check (qLDPC) codes defined on expander graphs.