  Nonlocal PNJL models allow for a detailed description of chiral quark
dynamics with running quark masses and wave function renormalization in
accordance with lattice QCD (LQCD) in vacuum. Their generalization to finite
temperature T and chemical potential \mu{} allows to reproduce the
\mu-dependence of the pseudocritical temperature from LQCD when a nonvanishing
vector meson coupling is adjusted. This restricts the region for the critical
endpoint in the QCD phase diagram and stiffens the quark matter equation of
state (EoS). It is demonstrated that the construction of a hybrid EoS for
compact star applications within a two-phase approach employing the nonlocal
PNJL EoS and an advanced hadronic EoS leads to the masquerade problem. A
density dependence of the vector meson coupling is suggested as a possible
solution which can be adjusted in a suitable way to describe hybrid stars with
a maximum mass in excess of 2 M_sun with a possible early onset of quark
deconfinement even in the cores of typical (M ~ 1.4 M_sun) neutron stars.
