  A phase of strong interacting matter with deconfined quarks is expected in
the core of massive neutron stars. We investigate the quark deconfinement phase
transition in cold (T = 0) and hot beta-stable hadronic matter. Assuming a
first order phase transition, we calculate and compare the nucleation rate and
the nucleation time due to quantum and thermal nucleation mechanisms. We show
that above a threshold value of the central pressure a pure hadronic star (HS)
(i.e. a compact star with no fraction of deconfined quark matter) is metastable
to the conversion to a quark star (QS) (i.e. a hybrid star or a strange star).
This process liberates an enormous amount of energy, of the order of
10^{53}~erg, which causes a powerful neutrino burst, likely accompanied by
intense gravitational waves emission, and possibly by a second delayed (with
respect to the supernova explosion forming the HS) explosion which could be the
energy source of a powerful gamma-ray burst (GRB). This stellar conversion
process populates the QS branch of compact stars, thus one has in the Universe
two coexisting families of compact stars: pure hadronic stars and quark stars.
We introduce the concept of critical mass M_{cr} for cold HSs and
proto-hadronic stars (PHSs), and the concept of limiting conversion temperature
for PHSs. We show that PHSs with a mass M < M_{cr} could survive the early
stages of their evolution without decaying to QSs. Finally, we discuss the
possible evolutionary paths of proto-hadronic stars.
