  A number of recent proposals for a quantum theory of gravity are based on the
idea that spacetime geometry and gravity are derivative concepts and only apply
at an approximate level. There are two fundamental challenges to any such
approach. At the conceptual level, there is a clash between the "timelessness"
of general relativity and emergence. Second, the lack of a fundamental
spacetime makes difficult the straightforward application of well-known methods
of statistical physics to the problem. We recently initiated a study of such
problems using spin systems based on evolution of quantum networks with no a
priori geometric notions as models for emergent geometry and gravity. In this
article we review two such models. The first is a model of emergent (flat)
space and matter and we show how to use methods from quantum information theory
to derive features such as speed of light from a non-geometric quantum system.
The second model exhibits interacting matter and geometry, with the geometry
defined by the behavior of matter. This model has primitive notions of
gravitational attraction which we illustrate with a toy black hole, and
exhibits entanglement between matter and geometry and thermalization of the
quantum geometry.
