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Recent experimental tests of Bell inequalities confirm that entangled quantumsystems cannot be described by local classical theories but still do not answerthe question whether or not quantum systems could in principle be modelled bylinear hidden variable theories. In this paper, we study the quantumtrajectories of a single qubit that experiences a sequence of repeatedgeneralised measurements. It is shown that this system, which constitutes aHidden Quantum Markov Model, is more likely to produce complex timecorrelations than any classical Hidden Markov Model with two output symbols.From this, we conclude that quantum physics cannot be replaced by linear hiddenvariable theories. Indeed, it has already been recognised that not onlyentanglement but also non-classical time correlations of quantum systems withquantum feedback are a valuable resource for quantum technology applications.

Quantum physics cannot be captured by classical linear hidden variable theories even in the absence of entanglement