Landau's idea of classifying phases of matter in terms of symmetry breaking is a cornerstone of modern physics. In his pioneering work Wilczek argued that an autonomous system can break time translation symmetry, thus realising what he named as time crystals. Their existence has been confirmed very recently in a number of experiments. I will briefly review the field focusing in particular on the so called Floquet time-crystals (arising in many-body systems that are periodically driven) and on time-crystalline behaviour in quantum open systems. For this last case, I will make connections to quantum synchronisation.

The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. I will describe results establishing the existence, and the geometric origin, of intrinsic sign problems in a broad class of topological phases in 2+1 dimensions. Within this class, these results exclude the possibility of 'stoquastic' Hamiltonians for bosons, and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The talk is based on Phys. Rev. Research 2, 043032 and 033515.