Challenges and opportunities in hot and dense QCD

The phase diagram for QCD is a function of the temperature, T, and the quark chemical potential, $mu$. Simulations of lattice QCD with classical computers have studied hot QCD, for $mu < T$, close to the continuum limit. I discuss how moving out in this plane, first to warm quark matter, and then to cold quark matter, is one of the major problems of physics in the 21st century. To treat this from first principles will require quantum computers, at some point in the hopefully not too distant future. Until then, theoretical efforts must rely upon effective models. These suggest that the phases of QCD are much richer at $mu eq 0$ than at $mu = 0$, and include a true critical (end)point, quarkyonic matter, a moat spectrum for pions and kaons, inhomogeneous (pion/kaon/quarkyonic) condensates or quantum pion/kaon liquids, and color superconductivity. Warm quark matter is measured experimentally through the collisions of heavy ions at intermediate energies, such as at the Beam Energy Scan at RHIC, and the Compressed Baryon Matter experiment at FAIR. Cold quark matter is directly relevant to neutron stars, and can be probed by the binary collisions of neutron stars with gravitational astronomy such as LIGO, and ultraviolet astronomy, as with the NICER experiment. The wealth of data which will emerge in the next decade represents an exciting challenge for theory.