  The origin of non-classicality in physical systems and its connection to
distinctly quantum features such as entanglement and coherence is a central
question in quantum physics. This work analyses this question theoretically and
experimentally, linking quantitatively non-classicality with quantum coherence.
On the theoretical front, we show when the coherence of an observable is
linearly related to the degree of violation of the Kolmogorov condition, which
quantifies the deviation from any classical (non-invasive) explanation of the
multi-time statistics. Experimentally, we probe this connection between
coherence and non-classicality in a time-multiplexed optical quantum walk. We
demonstrate exquisite control of quantum coherence of the walker by varying the
degree of coherent superposition effected by the coin, and we show a
concomitant variation in the degree of non-classicality of the walker
statistics, which can be accessed directly by virtue of the unprecedented
control on the measurement-induced effects obtained via fast programmable
electro-optic modulators.
