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.