  Liquid crystals are phases of matter intermediate between crystals and
liquids. Whereas classical liquid crystals have been known for a long time and
are used in electro-optical displays, much less is known about their quantum
counterparts. There is growing evidence that quantum liquid crystals play a
central role in many electron systems including high temperature
superconductors, but a quantitative understanding is lacking due to disorder
and other complications. Here, we analyse the quantum phase diagram of a
two-dimensional dipolar gas, which exhibits stripe, nematic and supersolid
phases. We calculate the stiffness constants determining the stability of the
nematic and stripe phases, and the melting of the stripes set by the
proliferation of topological defects is analysed microscopically. Our results
for the critical temperatures of these phases demonstrate that a controlled
study of the interplay between quantum liquid and superfluid phases is within
experimental reach for the first time, using dipolar gases.
