The process matrix formalism allows us to go beyond the quantum circuit framework and study processes for which there may be no global causal order between events. While the notion of causal (non)separability has been well studied in the bipartite scenario, multipartite scenarios present a variety of new phenomena. In such scenarios, the definition of causal separability itself is subtle; I will discuss the possible definitions of causal separability and present characterisations in terms of necessary and sufficient conditions.
Causally nonseparable processes that can be physically implemented – such as the quantum switch – are of particular practical interest. I will show how processes that satisfy the sufficient condition for causal separability can be interpreted as "quantum circuits with classical control of causal order". I will then introduce a natural generalistion: "quantum circuits with quantum control of causal order", and provide a characterisation of such processes. These processes are physically realisable and include not just the quantum switch, but also new types of causally separable processes when more parties are considered.