  Recently, table-top experiments involving massive quantum systems have been
proposed to test the interface of quantum theory and gravity. In particular,
the crucial point of the debate is whether it is possible to conclude anything
on the quantum nature of the gravitational field, provided that two quantum
systems become entangled solely due to the gravitational interaction.
Typically, this question has been addressed by assuming a specific physical
theory to describe the gravitational interaction, but no systematic approach to
characterise the set of possible gravitational theories which are compatible
with the observation of entanglement has been proposed. Here, we remedy this by
introducing the framework of Generalised Probabilistic Theories (GPTs) to the
study of the nature of the gravitational field. This framework enables us to
systematically study all theories compatible with the detection of entanglement
generated via the gravitational interaction between two systems. We prove a
no-go theorem stating that the following statements are incompatible: i)
gravity is able to generate entanglement; ii) gravity mediates the interaction
between the systems; iii) gravity is classical. We analyse the violation of
each condition, in particular with respect to alternative non-linear models
such as the Schr\"odinger-Newton equation and Collapse Models.

Quantum Physics;General Relativity and Quantum Cosmology