We combine results from astrophysical observations and theoretical ab-initio calculations to investigate the presence of quark matter ( QM ) inside neutron stars ( NSs ) .
We find a clear qualitative change in the material properties of NS matter at energy densities comparable to those where quark-gluon plasma is created in heavy-ion collisions .
In the low-density phase , the system has characteristics closely resembling those of hadronic matter , while the high-density phase can be clearly identified with nearly conformal QM .
We show that QM is never present inside 1.44 M _ { \odot } NSs , but resides inside maximally massive NSs barring very specific and extreme conditions .
Finally , for the heaviest observed NSs with M \approx 2 M _ { \odot } , the fate of QM is found to strongly depend on the behavior of the speed of sound in the matter : if this quantity does not strongly violate the conformal limit c _ { s } ^ { 2 } \leq 1 / 3 , these stars host sizable quark cores of R \gtrsim 5 km .