  We study properties of the recently established refined topological recursion
for some simple spectral curves associated to quadratic differentials. We prove
explicit formulas for the free energy and Voros coefficients of the
corresponding quantum curves, and conjecture expressions for all other (smooth)
genus zero degree two curves. The results can be written in terms of
Bridgeland's notion of refined BPS structure associated to the same initial
data, together with a quantum correction to the central charge. The
corresponding "invariants" appear to be new, but their interpretation in terms
of Donaldson-Thomas theory or QFT is not entirely clear.
