We take causal structure to be some relation defined over elements which are understood to be morphisms of some category. An example of such a relation is a domain, another is a directed acyclic graph. Yet another is a string diagram in a symmetric monoidal category.
Panangaden and Martin showed that interval domains are categorically equivalent to (hyperbolic?) spacetimes. This makes the domain a perfect candidated for classical relativistic causality. The planar graphs of the diagrammatical calculus are an enticing candidated for quantum causal structure. Then there is Hardy's causaloid. The classical causal structure seems very realist, as in we might believe in the existence of the set of events for a universe. I have offered the Fischer Impossibility result as a refutation of the naïve existence of this set.
What are the best candidates for a quantum causal structure?