  Determination of the quantum nature of correlations between two spatially
separated systems plays a crucial role in quantum information science. Of
particular interest is the questions of if and how these correlations enable
quantum information protocols to be more powerful. Here, we report on a
distributed quantum computation protocol in which the input and output quantum
states are considered to be classically correlated in quantum informatics.
Nevertheless, we show that the correlations between the outcomes of the
measurements on the output state cannot be efficiently simulated using
classical algorithms. Crucially, at the same time, local measurement outcomes
can be efficiently simulated on classical computers. We show that the only
known classicality criterion violated by the input and output states in our
protocol is the one used in quantum optics, namely, phase-space
nonclassicality. As a result, we argue that the global phase-space
nonclassicality inherent within the output state of our protocol represents
true quantum correlations.
