  Formulation of physical theories typically assumes a definite causal
structure -- either static or dynamic -- among the set of physical events.
Recent studies, however, suggest the possibility of indefiniteness in causal
structure, which emerges as a novel information primitive offering advantages
in various protocols. In this work, we explore utilities of this new primitive
in cryptographic applications. To this aim, we propose a task called
local-data-hiding, where a referee distributes encrypted messages among distant
parties in such a way that the parties individually remain completely ignorant
about the messages, and thus try to decrypt their respective messages through
mutual collaboration. As we demonstrate, agents embedded in an indefinite
causal structure can outperform their counterparts operating in a definite
causal background. Considering the bipartite local-bit-hiding (LBH) task, we
establish a strict duality between its optimal success probability and the
optimal violation of a causal inequality obtained from the
guess-your-neighbour's-input game. This, in turn, provides a way forward to
obtain Tsirelson-type bounds for causal inequalities. Furthermore, similar to
Peres's separability criterion, we derive a necessary criterion for quantum
processes to be useful in the LBH task. We then report an intriguing
super-activation phenomenon, where two quantum processes, each individually not
useful for the LBH task, become useful when used together. We also analyze the
utility of causal indefiniteness arising in classical setups and show its
advantages in multipartite variants of the local-data-hiding task. Along with
establishing new cryptographic applications our study illuminates various
unexplored aspects of causal indefiniteness, and welcomes further studies on
this new information primitive.
