  The constraints arising for a general set of causal relations, both
classically and quantumly, are still poorly understood. As a step in exploring
this question, we consider a coherently controlled superposition of
"direct-cause" and "common-cause" relationships between two events. We propose
an implementation involving the spatial superposition of a mass and general
relativistic time dilation. Finally, we develop a computationally efficient
method to distinguish such genuinely quantum causal structures from classical
(incoherent) mixtures of causal structures and show how to design experimental
verifications of the nonclassicality of a causal structure.
