  The chemical evolution of the Universe and several phases of the stellar life
are regulated by minute nuclear reactions. The key point for each of these
reactions is the value of cross sections at the energies at which they take
place in stellar environments. Direct cross-section measurements are mainly
hampered by the very low counting rate and by cosmic background, nevertheless
they have become possible by combining the best experimental techniques with
the cosmic silence of an underground laboratory. In the nineties the LUNA
(Laboratory for Underground Nuclear Astrophysics) collaboration opened the era
of underground nuclear astrophysics installing first a home-made 50 kV and
later on, a second 400 kV accelerator under the Gran Sasso mountain in Italy:
in 25 years of experimental activity, important reactions responsible for
hydrogen burning could have been studied down to the relevant energies thanks
to the high current proton and helium beams provided by the machines. The
interest to the next and warmer stages of star evolution (i.e. post main
sequence, helium and carbon burning) drove a new project based on an ion
accelerator in the MV range called LUNA-MV, able to deliver proton, helium and
carbon beams. The present contribution is aimed to discuss the state-of-the-art
for some selected key processes of post main sequence stellar phases: the 12Cag
and the 12C+12C fundamental for helium and carbon burning phases, and 13Can and
22Neag that are relevant to the synthesis of heavy elements in AGB stars. The
perspectives opened by an underground MV-facility will be highlighted.
