In recent years we've expanded our understanding of entanglement in many-body quantum systems; both how it behaves in ground states, and how it grows out-of-equilibrium. Entanglement is very difficult to measure in experiments. But through understanding it better, we've made great progress in classifying quantum phases of matter, and in developing algorithms for efficiently simulating quantum systems. I will review some recent progress in these directions.

In recent years we've greatly expanded our understanding of entanglement in many-body quantum systems; both how it behaves in ground states, and how it grows out-of-equilibrium. While entanglement is very difficult to measure in experiments, it has nevertheless driven progress in 1) the classification of quantum phases of matter and 2) strategies for efficiently simulating many-body systems on classical and quantum computers. I will review some recent progress in these directions. Along the way I will summarise some older results on how entanglement grows in many-body systems, briefly highlight some connections to holography, and present a conjecture about the asymptotic computational difficulty of calculating transport in many-body systems.