  We report on a stringent test of the non-classicality of the motion of a
massive quantum particle, which propagates on a discrete lattice. Measuring
temporal correlations of the position of single atoms performing a quantum
walk, we observe a $6\sigma$ violation of the Leggett-Garg inequality. Our
results rigorously excludes (i.e. falsifies) any explanation of quantum
transport based on classical, well-defined trajectories. We use so-called ideal
negative measurements -- an essential requisite for any genuine Leggett-Garg
test -- to acquire information about the atom's position, yet avoiding any
direct interaction with it. The interaction-free measurement is based on a
novel atom transport system, which allows us to directly probe the absence
rather than the presence of atoms at a chosen lattice site. Beyond the
fundamental aspect of this test, we demonstrate the application of the
Leggett-Garg correlation function as a witness of quantum superposition. We
here employ the witness to discriminate different types of walks spanning from
merely classical to wholly quantum dynamics.
