The theory governing the strong nuclear force—quantum chromodynamics—predicts that at sufficiently high energy densities, hadronic nuclear matter undergoes a deconfinement transition to a new phase of quarks and gluons1. Although this has been observed in ultrarelativistic heavy-ion collisions2,3, it is currently an open question whether quark matter exists inside neutron stars4. By combining astrophysical observations and theoretical ab initio calculations in a model-independent way, we find that the inferred properties of matter in the cores of neutron stars with mass corresponding to 1.4 solar masses (M⊙) are compatible with nuclear model calculations. However, the matter in the interior of maximally massive stable neutron stars exhibits characteristics of the deconfined phase, which we interpret as evidence for the presence of quark-matter cores. For the heaviest reliably observed neutron stars5,6 with mass M ≈ 2M⊙, the presence of quark matter is found to be linked to the behaviour of the speed of sound cs in strongly interacting matter. If the conformal bound $${c}_{\rm{s}}^{2}\le 1/3$$ c s 2 ≤ 1 / 3 (ref. 7) is not strongly violated, massive neutron stars are predicted to have sizable quark-matter cores. This finding has important implications for the phenomenology of neutron stars and affects the dynamics of neutron star mergers with at least one sufficiently massive participant.

The cooling phase of thermonuclear (type-I) X-ray bursts can be used to constrain the neutron star (NS) compactness by comparing the observed cooling tracks of bursts to accurate theoretical atmosphere model calculations. By applying the so-called cooling tail method, where the information from the whole cooling track is used, we constrain the mass, radius, and distance for three different NSs in low-mass X-ray binaries 4U 1702−429, 4U 1724−307, and SAX J1810.8−260. Care is taken to only use the hard state bursts where it is thought that only the NS surface alone is emitting. We then utilize a Markov chain Monte Carlo algorithm within a Bayesian framework to obtain a parameterized equation of state (EoS) of cold dense matter from our initial mass and radius constraints. This allows us to set limits on various nuclear parameters and to constrain an empirical pressure-density relation for the dense matter. Our predicted EoS results in NS radius between 10.5 − 12.8 km (95% confidence limits) for a mass of 1.4 M depending slightly on the assumed composition. Due to systematic errors and uncertainty in the composition these results should be interpreted as lower limits for the radius.