  Many features of single-partite quantum walks can be simulated by classical
waves. However, it was recently experimentally shown that some temporal
sequences of measurements on a quantum walker do not admit a classical
description in terms of macro-realistic theories. Here, we extend this
observation and present pre- and post-selection scenarios in quantum walks
leading to logical paradoxes. Such paradoxes were recently shown to be
equivalent to proofs of contextuality, therefore we provide an additional
argument for non-classicality of a quantum walk model. The assumptions behind
the claims of non-classicality (including contextuality and violations of
macro-realism) are also discussed.
