  Whether or not deconfined quark phase exists in neutron star cores and
represents the ground state of neutral matter at moderate densities are open
questions. We use two realistic effective quark models, the three-flavor
Nambu-Jona-Lasinio model and the modified quark-meson coupling model, to
describe the neutron star matter. After constructing possible hybrid equations
of state (EOSes) with unpaired or color superconducting quark phase, we
systematically discuss the observational constraints of neutron stars on the
EOSes. It is found that the neutron star with pure quark matter core is
unstable and the hadronic phase with hyperons is denied, while hybrid EOSes
with two-flavor color superconducting phase or unpaired quark matter phase are
both allowed by the tight and most reliable constraints from two stars Ter 5 I
and EXO 0748-676. And the hybrid EOS with unpaired quark matter phase is
allowed even compared with the tightest constraint from the most massive pulsar
star PSR J0751+1807. Therefore, we conclude that the ground state of neutral
matter at moderate densities is in deconfined quark phase likely.
