  We discuss the Cardy limit of 3d supersymmetric partition functions which
allow the factorization into the hemisphere indices: the generalized
superconformal index, the refined topologically twisted index and the squashed
sphere partition function. In the Cardy limit, the hemisphere index can be
evaluated by the saddle point approximation where there exists a dominant
saddle point contribution, which we call the Cardy block. The Cardy block turns
out to be a simple but powerful object as it is a building block of other
partition functions in the Cardy limit. The factorization to the Cardy block
allows us to find universal relations among the partition functions, which we
formulate as index theorems. Furthermore, if we consider a holographic 3d SCFT
and its large $N$ limit, those partition functions relate to various entropic
quantities of the dual gravity theory in AdS$_4$. As a result, our result
provides the microscopic derivation of the universal relations among those
entropic quantities of the gravity theory. We also discuss explicit examples,
which confirm our general index theorems.
