  We propose and test exact quantization conditions for the $N$-particle
quantum elliptic Ruijsenaars-Schneider integrable system, as well as its
Calogero-Moser limit, based on the conjectural correspondence to the
five-dimensional $\mathcal{N} = 1$ $SU(N)$ gauge theory in the
Nekrasov-Shatashvili limit. We discuss two natural sets of quantization
conditions, related by the electro-magnetic duality, and the importance of
non-perturbative corrections in the Planck constant. We also comment on the
eigenfunction problem, by reinterpreting the Separation of Variables approach
in gauge theory terms.
