Equation of state in neutron stars and supernovae

Neutron stars and supernovae provide cosmic laboratories of highly compressed matter at supra nuclear saturation density which is beyond the reach of terrestrial experiments. The properties of dense matter is extracted by combining the knowledge of nuclear experiments and astrophysical observations via theoretical frameworks. A matter in neutron stars is neutron rich, and may further accommodate non-nucleonic degrees of freedom such as hyperons and quarks. The structure and composition of neutron stars are determined by equations of state of matter, which are the primary subject in this chapter. In case of supernovae, the time evolution includes several dynamical stages whose descriptions require equations of state at finite temperature and various lepton fractions. Equations of state also play essential roles in neutron star mergers which allow us to explore new conditions of matter not achievable in static neutron stars and supernovae. Several types of hadron-to-quark transitions, from first order transitions to crossover, are reviewed, and their characteristics are summarized.