  We argue that the full conversion of a hadronic star into a quark or a hybrid
star occurs within two different regimes separated by a critical value of the
density of the hadronic phase $\overline{n_h}$. The first stage, occurring for
$n_h>\overline{n_h}$, is characterized by turbulent combustion and lasts
typically a few ms. During this short time-scale neutrino cooling is basically
inactive and the star heats up thanks to the heat released in the conversion.
In the second stage, occurring for $n_h<\overline{n_h}$, turbulence is not
active anymore, and the conversion proceeds on a much longer time scale (of the
order of tens of seconds), with a velocity regulated by the diffusion and the
production of strange quarks. At the same time, neutrino cooling is also
active. The interplay between the heating of the star due to the slow
conversion of its outer layers (with densities smaller than $\overline{n_h}$)
and the neutrino cooling of the forming quark star leads to a quasi-plateau in
the neutrino luminosity which, if observed, would possibly represent a unique
signature for the existence of quark matter inside compact stars. We will
discuss the phenomenological implications of this scenario in particular in
connection with the time structure of long gamma-ray-bursts.
