  Figuring out the physical rationale behind natural selection of quantum
theory is one of the most acclaimed quests in quantum foundational research.
This pursuit has inspired several axiomatic initiatives to derive mathematical
formulation of the theory by identifying general structure of state and effect
space of individual systems as well as specifying their composition rules. This
generic framework can allow several consistent composition rules for a
multipartite system even when state and effect cones of individual subsystems
are assumed to be quantum. Nevertheless, for any bipartite system, none of
these compositions allows beyond quantum space-like correlations. In this
letter we show that such bipartite compositions can admit stronger than quantum
correlations in the time-like domain and, hence, indicates pragmatically
distinct roles carried out by state and effect cones. We discuss consequences
of such correlations in a communication task, which accordingly opens up a
possibility of testing the actual composition between elementary quanta.
