  The concept of causal nonseparability has been recently introduced, in
opposition to that of causal separability, to qualify physical processes that
locally abide by the laws of quantum theory, but cannot be embedded in a
well-defined global causal structure. While the definition is unambiguous in
the bipartite case, its generalisation to the multipartite case is not so
straightforward. Two seemingly different generalisations have been proposed,
one for a restricted tripartite scenario and one for the general multipartite
case. Here we compare the two, showing that they are in fact inequivalent. We
propose our own definition of causal (non)separability for the general case,
which---although a priori subtly different---turns out to be equivalent to the
concept of "extensible causal (non)separability" introduced before, and which
we argue is a more natural definition for general multipartite scenarios. We
then derive necessary, as well as sufficient conditions to characterise
causally (non)separable processes in practice. These allow one to devise
practical tests, by generalising the tool of witnesses of causal
nonseparability.
