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Given a unitary evolution $U$ on a multi-partite quantum system and anensemble of initial states, how well can $U$ be simulated by local operationsand classical communication (LOCC) on that ensemble? We answer this question byestablishing a general, efficiently computable upper bound on the maximal LOCCsimulation fidelity -- what we call an "LOCC inequality". We then apply ourfindings to the fundamental setting where $U$ implements a quantum NewtonianHamiltonian over a gravitationally interacting system. Violation of our LOCCinequality can rule out the LOCCness of the underlying evolution, therebyestablishing the non-classicality of the gravitational dynamics, which can nolonger be explained by a local classical field. As a prominent application ofthis scheme we study systems of quantum harmonic oscillators initialised incoherent states following a normal distribution and interacting via Newtoniangravity, and discuss a possible physical implementation with torsion pendula.One of our main technical contributions is the analytical calculation of theabove LOCC inequality for this family of systems. As opposed to existing testsbased on the detection of gravitationally mediated entanglement, our proposalworks with coherent states alone, and thus it does not require the generationof largely delocalised states of motion nor the detection of entanglement,which is never created at any point in the process.

Testing the quantum nature of gravity without entanglement