  Hyperon ($Y$) mixing in neutron-star matter brings about a remarkable
softening of the equation of state (EoS) and the maximum mass is reduced to a
value far less than $2M_{\odot}$. One idea to avoid this "hyperon puzzle in
neutron stars" is to assume that the many-body repulsions work universally for
every kind of baryons. The other is to take into account the quark
deconfinement phase transitions from a hadronic EoS to a sufficiently stiff
quark-matter EoS. In the present approach, both effects are handled in a common
framework. As well as the hadronic matter, the quark matter with the two-body
quark-quark interactions are treated within the Brueckner-Bethe-Goldstone
theory beyond the mean field frameworks, where interaction parameters are based
on the terrestrial data. The derived mass-radius relations of neutron stars
show that maximum masses reach over $2M_{\odot}$ even in the cases of including
hadron-quark phase transitions, being consistent with the recent observations
for maximum masses and radii of neutron stars by the NICER measurements and the
other multimessenger data.
