  We consider a large class of $q$-series that have the structure of Nahm sums,
or equivalently motivic generating series for quivers. First, we initiate a
systematic analysis and classification of classical and quantum A-polynomials
associated to such $q$-series. These quantum quiver A-polynomials encode
recursion relations satisfied by the above series, while classical
A-polynomials encode asymptotic expansion of those series. Second, we postulate
that those series, as well as their quantum quiver A-polynomials, can be
reconstructed by means of the topological recursion. There is a large class of
interesting quiver A-polynomials of genus zero, and for a number of them we
confirm the above conjecture by explicit calculations. In view of recently
found dualities, for an appropriate choice of quivers, these results have a
direct interpretation in topological string theory, knot theory, counting of
lattice paths, and related topics. In particular it follows, that various
quantities characterizing those systems, such as motivic Donaldson-Thomas
invariants, various knot invariants, etc., have the structure compatible with
the topological recursion and can be reconstructed by its means.
