The final report of grant FA9550-16-1-0256 details the developments and successes made in the exponentially-complex computational challenge posed by quantum many-body problems throughout chemistry, materials science and condensed matter fields of research. The work focused on the development of fast randomized iteration approaches for computing thermal and response properties of systems in these fields, via novel stochastic sampling methods, which enabled beyond state-of-the-art calculations on a number of key correlated electron systems. These included the polarizability of molecular systems, spin correlations of graphene sheets, and thermal calculations on strongly correlated lattice models. In addition to thermal and response properties, the ability to stochastically project into appropriate subspaces was considered, and the limitations on such an approach detailed. As an alternate stochastic compression, the work enabled by this grant has also investigated a novel approach to stochastically sampling a compressed, non-linear parameterization of the solution vector. This work draws on inspiration from the optimization of neural networks in order to hugely increase the complexity of states which can be considered in this way. Key successes of the work include the development of a new algorithm to stochastically sample the response of a quantum system to perturbations, the demonstration of this algorithm to resolve a discrepancy in the literature based on alternative (inexact) methods, the development of a new computational approach to sample and optimize highly flexible and complex non-linear wave function ansatze, and the thorough analysis of the limitation of stochastic subspace sampling, which has found use in some proposed quantum computing algorithms.