At low temperatures non-equilibrium voltage fluctuations can be generated in current-biased superconducting nanowires due to proliferation of quantum phase slips (QPS) or, equivalently, due to quantum tunneling of magnetic flux quanta across the wire. In this paper we review and further extend recent theoretical results related to this phenomenon. Employing the phase-charge duality arguments combined with Keldysh path integral technique we analyze such fluctuations within the two-point and four-point measurement schemes demonstrating that voltage noise detected in such nanowires in general depends on the particular measurement setup. In the low frequency limit we evaluate all cumulants of the voltage operator which turn out to obey Poisson statistics and exhibit a power law dependence on the external bias. We also specifically address a non-trivial frequency dependence of quantum shot noise power spectrum <$E S sub OMEGA> for both longer and shorter superconducting nanowires. In particular, we demonstrate that <$E S sub OMEGA> decreases with increasing frequency <$E OMEGA> and vanishes beyond a threshold value of <$E OMEGA> at <$E T~symbol О~0>. Furthermore, we predict that <$E S sub OMEGA> may depend non-monotonously on temperature due to quantum coherent nature of QPS noise. The results of our theoretical analysis can be directly tested in future experiments with superconducting nanowires.

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