X-ray observations of quiescent low-mass X-ray binaries (qLMXBs) provide one of the methods to understand the internal structure of neutron stars and therefore place constraints on the nuclear physics of dense matter. The hot thermal emission from the surface of neutron stars in qLMXBs permits measurements of the neutron star radius. In the past few years, promising results were obtained from statistical analyses that combined the X-ray spectra of qLMXBs. These analyses have required simplistic or analytical parametrizations of the Equation of State of dense matter (the relation between pressure and density of matter). In this talk, I will summarize these previous results, and present new constraints obtained from the direct confrontation between a realistic parametrization of the equation of state and the astrophysical data from qLMXBs In an effort to be conservative with the observational constraints, I will also discuss the current limitations of the qLMXB method, and how they can be overcome with more observations of qLXMBs with current and future instrumentation.