  In the first part of this paper, we investigate the possible existence of a
structured hadron-quark mixed phase in the cores of neutron stars. This phase,
referred to as the hadron-quark pasta phase, consists of spherical blob, rod,
and slab rare phase geometries. Particular emphasis is given to modeling the
size othis phase in rotating neutron stars. We use the relativistic mean-field
theory to model hadronic matter and the non-local three-flavor
Nambu-Jona-Lasinio model to describe quark matter. Based on these models, the
hadron-quark pasta phase exists only in very massive neutron stars, whose
rotational frequencies are less than around 300 Hz. All other stars are not
dense enough to trigger quark deconfinement in their cores. Part two of the
paper deals with the quark-hadron composition of hot (proto) neutron star
matter. To this end we use a local three-flavor Polyakov-Nambu-Jona-Lasinio
model which includes the 't Hooft (quark flavor mixing) term. It is found that
this term leads to non-negligible changes in the particle composition of
(proto) neutron stars made of hadron-quark matter.
