  We analyze the quantum melting of two-dimensional Wigner molecules (WM) in
confined geometries with distinct symmetries and compare it with corresponding
thermal melting. Our findings unfold complementary mechanisms that drive the
quantum and thermal crossovers in a WM and show that the symmetry of the
confinement plays no significant role in determining the quantum crossover
scale $n_X$. This is because the zero-point motion screens the boundary effects
within short distances. The phase diagram as a function of thermal and quantum
fluctuations determined from independent criteria is unique, and shows
"melting" from the WM to both the classical and quantum "liquids." An
intriguing signature of weakening liquidity with increasing temperature, $T$,
is found in the extreme quantum regime. The crossover is associated with
production of defects. However, these defects appear to play distinct roles in
driving the quantum and thermal "melting." Our study will help comprehending
melting in a variety of experimental traps - from quantum dots to complex
plasma.
