The talk will illustrate some recent progress in enumerating the microstates of 1/16 BPS black holes in AdS5. Through holography, a generating function for this counting problem is the supersymmetric index of the dual SCFT. Imposing non-trivial boundary conditions for the SCFT fields around a circle in the geometry and taking the circumference to be small allows to set up a supersymmetric 3d effective field theory which captures the black hole microstates. I will describe how, up to exponentially small corrections, the index is then completely characterized by a universal, finite series made of 3d local terms, the supersymmetric Casimir energy and a simple logarithmic term. Taking the large-N limit of this series reproduces the Bekenstein-Hawking black hole entropy. The subleading terms in the large-N expansion are expected to match higher-derivative and quantum corrections to the entropy. The last part of the talk will discuss current efforts towards computing such corrections directly on the gravity side.