Confirming or ruling out the existence of deconfined quark matter inside neutron stars is one of the most prominent open problems in nuclear astrophysics. While the ultimate goal continues to be the observation of a smoking gun signal directly indicating the presence or creation of quark matter, a more indirect approach to the problem has lately become feasible. By combining ab-initio theoretical results for the microscopic properties of dense QCD matter with the latest astrophysical measurements of neutron star properties, it is possible to build stringent model-independent constraints for the material properties of neutron-star matter at different densities. I will present recent results concerning both a perturbative determination of the equation of state of quark matter and a model-independent study of quark-matter cores in massive neutron stars. The results of both studies indicate that matter in the cores of the heaviest stable neutron stars has characteristics much closer to deconfined quark matter than low-density nuclear matter.