  In this paper we describe algebraic and diagrammatic methods, related to the
MHV generating function method, for evaluating and exposing the structure of
supersymmetric sums over the states crossing generalized unitarity cuts of
multi-loop amplitudes in four dimensions. We focus mainly on cuts of maximally
supersymmetric Yang-Mills amplitudes. We provide various concrete examples,
some of which are directly relevant for the calculation of four-loop
amplitudes. Additionally, we discuss some cases with less than maximal
supersymmetry. The results of these constructions carry over to generalized
cuts of multi-loop supergravity amplitudes through use of the
Kawai-Lewellen-Tye relations between gravity and gauge-theory tree amplitudes.
