Hi Matt,
I have not tried to get the quantum-mechanical aspect of it, but I have done some work on using this to describe discrete Relativistic structures. So think of this as in the spirit of causal sets. I can get a simple discrete approximation to a 2-D Minkowski space-time and to a 3-D inflating space-time with horizons, and this is just from trying a few simple constructive rules for the Linear Structure and then analyzing the results. I have an idea for a general scheme for writing down constructive rules (both deterministic and stochastic) for generating Relativistic discrete Linear Structures, but there is a lot of work to do.
Just to give a taste of how this differs from causal sets, using the usual way that causal sets are generated no pair of events will be null related. But doing it my way, the entire space-time structure is built from null related events: it is all light-like in the foundations, as it were. I can also easily put in place constraints on the constructive rules that avoid some of the issues that come up for causal sets, which basically arise from the fact that the kind of graph they want to get is very much not a random graph.
The analytical advantage of a discrete space is that it comes already equipped with a natural measure--counting measure--but in the Relativistic case you have to be careful about what to count. I know that sounds cryptic, but it would take to long to explain properly...maybe we can talk about it sometime.
It may be that just being able to generate good discrete approximations to classical solutions in GR would yield clues about how to implement a fully quantum treatment, but that prospect is too far away now.
Cheers,
Tim