Quark stars have some special characteristics that separate them from ordinary neutron stars. Under the physical conditions found inside neutron stars, with extremely high densities but temperatures well below 10 K, quark matter is predicted to exhibit some peculiar characteristics. It is expected to behave as a Fermi liquid and enter a so-called color-flavor-locked (CFL) phase of color superconductivity, where "color" refers to the six "charges" exhibited in the strong interaction, instead of the two charges (positive and negative) in electromagnetism. At slightly lower densities, corresponding to higher layers closer to the surface of the compact star, the quark matter will behave as a non-CFL quark liquid, a phase that is even more mysterious than CFL and might include color conductivity and/or several additional yet undiscovered phases. None of these extreme conditions can currently be recreated in laboratories so nothing can be inferred about these phases from direct experiments.