  We present a formula for the quantum entropy of supersymmetric
five-dimensional spinning black holes in M-theory compactified on $CY_3$, i.e.,
BMPV black holes. We use supersymmetric localization in the framework of
off-shell five dimensional $N=2$ supergravity coupled to $I = 1,\dots,N_V + 1$
off-shell vector multiplets. The theory is governed at two-derivative level by
the symmetric tensor $\mathcal{C}_{IJK}$ (the intersection numbers of the
Calabi-Yau) and at four-derivative level by the gauge-gravitational
Chern-Simons coupling $c_I$ (the second Chern class of the Calabi-Yau). The
quantum entropy is an $N_V + 2$-dimensional integral parameterised by one real
parameter $\varphi^I$ for each vector multiplet and an additional parameter
$\varphi^0$ for the gravity multiplet. The integrand consists of an action
governed completely by $\mathcal{C}_{IJK}$ and $c_{I}$, and a one-loop
determinant. Consistency with the on-shell logarithmic corrections to the
entropy, the symmetries of the very special geometry of the moduli space, and
an assumption of analyticity constrains the one-loop determinant up to a
scale-independent function $f(\varphi^0)$. For $f=1$ our result agrees
completely with the topological M-theory conjecture of Dijkgraaf, Gukov,
Nietzke, and Vafa for static black holes at two derivative level, and provides
a natural extension to higher derivative corrections. For rotating BMPV black
holes, our result differs from the DGNV conjecture at the level of the first
quantum corrections.
