  We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the
number of affine (resp. linear) equivalence classes of, by increasing order of
generality, 2-level d-polytopes, d-cones and d-configurations. This in
particular answers positively a conjecture of Bohn et al. on 2-level polytopes.
We obtain our upper bound by relating affine (resp. linear) equivalence classes
of 2-level d-polytopes, d-cones and d-configurations to faces of the
correlation cone. We complement this with a $2^{\Omega(d^2)}$ lower bound, by
estimating the number of nonequivalent stable set polytopes of bipartite
graphs.
