  QCD with a finite baryon chemical potential, despite its importance, is not
well understood because the standard lattice QCD simulation is not applicable
due to the sign problem. Although QCD-like theories which do not suffer from
the sign problem have been studied intensively, relation to QCD with a finite
baryon chemical potential was not clear. This paper introduces large-$N_c$
equivalences between QCD and various QCD-like theories. These equivalences lead
us to a unified viewpoint for QCD with baryon and isospin chemical potentials,
$SO(2N_c)$ and $Sp(2N_c)$ gauge theories, QCD with adjoint matters and
two-color QCD. In particular QCD with the baryon chemical potential is
large-$N_c$ equivalent to its phase quenched version in a certain parameter
region, which is relevant for heavy ion collision experiments. All previous
simulation results which study the effect of the phase confirm the phase quench
approximation is quantitatively good already at Nc=3; it is so good that often
two theories give the same value within error. Therefore the phase quenched
simulation is the best strategy for the QCD critical point search. At small
volume one can study a tiny 1/Nc effect by the phase reweighting; the large-Nc
equivalence guarantees that the phase reweighing method works without suffering
from the overlapping problem.
