We discuss possible scenarios for the existence of strange matter in compact stars. The appearance of hyperons leads to a hyperon puzzle in ab-initio approaches based on effective baryon-baryon potentials but is not a severe problem in relativistic mean field models. In general, the puzzle can be resolved in a natural way if hadronic matter gets stiffened at supersaturation densities, an effect based on the quark Pauli quenching between hadrons. We explain the conflict between the necessity to implement dynamical chiral symmetry breaking into a model description and the conditions for the appearance of absolutely stable strange quark matter that require both, approximately masslessness of quarks and a mechanism of confinement. The role of strangeness in compact stars (hadronic or quark matter realizations) remains unsettled. It is not excluded that strangeness plays no role in compact stars at all. To answer the question whether the case of absolutely stable strange quark matter can be excluded on theoretical grounds requires an understanding of dense matter that we have not yet reached.
Chiral Symmetry Breaking, Color Superconductivity, and Equation of State for Magnetized Strange Quark Matter
