Quantum theory cannot violate a causal inequality

Within quantum theory, we can create superpositions of different causal orders of events, and observe interference between them. This raises the question of whether quantum theory can produce results that would be impossible to replicate with any classical causal model, thereby violating a causal inequality. This would be a temporal analogue of Bell inequality violation, which proves that no local hidden variable model can replicate quantum results. However, unlike the case of non-locality, we show that quantum experiments can be simulated by a classical causal model, and therefore cannot violate a causal inequality.