  Compact stars with significant high densities in their interiors can give
rise to quark deconfined phases that can open a window for the study of
strongly interacting dense nuclear matter. Recent observations on the mass of
two pulsars, PSR J1614-2230 and PSR J0348+0432, have posed a great restriction
on their composition, since their equations of state must be hard enough to
support masses of about at least two solar masses. The onset of quarks tends to
soften the equation of state, but due to their strong interactions, different
phases can be realized with new parameters that affect the corresponding
equations of state and ultimately the mass-radius relationships. In this paper
I will review how the equation of state of dense quark matter is affected by
the physical characteristics of the phases that can take place at different
baryonic densities with and without the presence of a magnetic field, as well
as their connection with the corresponding mass-radius relationship, which can
be derived by using different models.
