  In high-energy astrophysical processes involving compact objects, such as
core-collapse supernovae or binary neutron star mergers, neutrinos play an
important role in the synthesis of nuclides. Neutrinos in these environments
can experience collective flavor oscillations driven by neutrino-neutrino
interactions, including coherent forward scattering and incoherent
(collisional) effects. Recently, there has been interest in exploring potential
novel behaviors in collective oscillations of neutrinos by going beyond the
one-particle effective or "mean-field" treatments. Here, we seek to explore
implications of collective neutrino oscillations, in the mean-field treatment
and beyond, for the nucleosynthesis yields in supernova environments with
different astrophysical conditions and neutrino inputs. We find that collective
oscillations can impact the operation of the $\nu p$-process and $r$-process
nucleosynthesis in supernovae. The potential impact is particularly strong in
high-entropy, proton-rich conditions, where we find that neutrino interactions
can nudge an initial $\nu p$ process neutron rich, resulting in a unique
combination of proton-rich low-mass nuclei as well as neutron-rich high-mass
nuclei. We describe this neutrino-induced neutron capture process as the
"$\nu$i process". In addition, nontrivial quantum correlations among neutrinos,
if present significantly, could lead to different nuclide yields compared to
the corresponding mean-field oscillation treatments, by virtue of modifying the
evolution of the relevant one-body neutrino observables.
