  Quantum theory is in principle compatible with processes that violate causal
inequalities, an analogue of Bell inequalities that constrain the correlations
observed by a set of parties operating in a definite causal order. Since the
introduction of causal inequalities, determining their maximum quantum
violation, analogue to Tsirelson's bound for Bell inequalities, has remained an
open problem. Here we provide a general method for bounding the violation of
arbitrary causal inequalities, establishing limits on the correlations
achievable by arbitrary local experiments and by arbitrary quantum processes
with indefinite causal order. We prove that the maximum violation is generally
smaller than the algebraic maximum, and determine Tsirelson-like bounds for a
class of causal inequalities including some of the most paradigmatic examples.
Surprisingly, we find that the algebraic maximum of arbitrary causal
inequalities can be achieved by a new type of processes that allow for
information to flow in an indefinite temporal direction within the parties'
laboratories.
