  We report on the state of the art of quantum walk experiments with neutral
atoms in state-dependent optical lattices. We demonstrate a novel
state-dependent transport technique enabling the control of two spin-selective
sublattices in a fully independent fashion. This transport technique allowed us
to carry out a test of single-particle quantum interference based on the
violation of the Leggett-Garg inequality and, more recently, to probe
two-particle quantum interference effects with neutral atoms cooled into the
motional ground state. These experiments lay the groundwork for the study of
discrete-time quantum walks of strongly interacting, indistinguishable
particles to demonstrate quantum cellular automata of neutral atoms.
