The quantum measurement problem would be solved were quantum mechanics to be discovered to be an approximation to a theory with a precise description of individual systems. Quantum mechanics would arise as the statistical mechanics of that theory. The challenge is that this theory must be non-local to satisfy the tests of the Bell inequalities. But remarkably, there is evidence from research in quantum gravity that space and locality are emergent. This suggests that the ultimate reality might be described by a non-local theory from which space and the quantum both emerge. The goal of this project is to construct such a theory. Earlier work has already shown that theories that are both pre-quantum and pre-geometric can be expressed in a common mathematical language, which is the dynamics of matrices and graphs. This project will also attempt to develop experimental signatures of the emergence of space and the quantum. If space is emergent then there is a scale, usually taken to be the planck scale, below which lorentz invariance breaks down or is modified. This is expected to lead to effects detectible in present observations by astrophysical observatories such as AUGER and Fermi. This is a high risk/high payoff project. If fully successful, the enduring impact will be the replacement of the current formulation of a quantum mechanics, with a deeper theory; this will also lead to the exploration of novel experimental phenomena.