  In the macroscopic world, time is intrinsically asymmetric, flowing in a
specific direction, from past to future. However, the same is not necessarily
true for quantum systems, as some quantum processes produce valid quantum
evolutions under time reversal. Supposing that such processes can be probed in
both time directions, we can also consider quantum processes probed in a
coherent superposition of forwards and backwards time directions. This yields a
broader class of quantum processes than the ones considered so far in the
literature, including those with indefinite causal order. In this work, we
demonstrate for the first time an operation belonging to this new class: the
quantum time flip. Using a photonic realisation of this operation, we apply it
to a game formulated as a discrimination task between two sets of operators.
This game not only serves as a witness of an indefinite time direction, but
also allows for a computational advantage over strategies using a fixed time
direction, and even those with an indefinite causal order.
