  Recent advances in cold atom experimentation suggest that studies of quantum
two-dimensional melting of dipolar molecules, with dipoles aligned
perpendicular to ordering plane, may be on the horizon. An intriguing aspect of
this problem is that two-dimensional \emph{classical} aligned dipoles (already
studied in great detail in soft matter experiments on magnetic colloids) are
known to melt via a two-stage process, with an intermediate hexatic phase
separating the usual crystal and isotropic fluid phases. We estimate here the
effect of quantum fluctuations on this hexatic phase, for both dipolar systems
and charged Wigner crystals. Our approximate phase diagrams rely on a pair of
Lindemann criteria, suitably adapted to deal with effects of thermal
fluctuations in two dimensions. As part of our analysis, we determine the
phonon spectra of quantum particles on a triangular lattice interacting with
repulsive $1/r^3$ and $1/r$ potentials. A large softening of the transverse and
longitudinal phonon frequencies, due to both lattice effects and quantum
fluctuations, plays a significant role in our analysis. The hexatic phase is
predicted to survive down to very low temperatures.
