  In plethora of physical situations one can distinguish a mediator -- a system
that couples other, non-interacting systems. Often the mediator itself is not
directly accessible to experimentation, yet it is interesting and sometimes
crucial to understand if it admits non-classical properties. An example of this
sort that recently enjoys considerable attention are two quantum masses coupled
via gravitational field. It has been argued that the gain of quantum
entanglement between the masses indicates non-classicality of the states of the
whole tripartite system. Here, we focus on non-classical properties of the
involved interactions rather than the states. We derive inequalities whose
violation indicates non-commutativity and non-decomposability (open system
generalisation of non-commuting unitaries) of interactions through the
mediators. The derivations are based on properties of general quantum formalism
and make minimalistic assumptions about the studied systems, in particular the
interactions can remain uncharacterised throughout the assessment. Furthermore,
we also present conditions that solely use correlations between the coupled
systems, excluding the need to measure the mediator. Next, we show that the
amount of violation places a lower bound on suitably defined degree of
non-decomposability. This makes the methods quantitative and at the same time
experiment ready. We give applications of these techniques in two different
fields: for detecting non-classicality of gravitational interaction and in
bounding the Trotter error in quantum simulations.
