Toy model for quantum causality in finite topological spaces using colouring rules

In the quest for quantum gravity, Hardy suggests the radical aspects of quantum physics (probabilistic nature) and relativity (dynamic causality) would manifest together. In recent years, this has led to an active study of causally neutral formulations and more particularly the study of indefinite causal structures, that allow for perplexing phenomenon such as the quantum switch. In this work, I construct a toy model set in finite topological spaces with 1+1 D that have indefinite causal structures emerging from local definite causal orders through graph colouring rules and simple operational rules. In the setting of this toy model, I study indefinite causal structures that emerge from these operational rules and that naturally forbid close-time-like-curves. I study a version of the quantum switch and more such curious possibilities in this toy model and attempt to interpret these. The hope is to help shed some light on the study of indefinite causal structures. Future work will involve generalizing these results to higher dimensions.