We characterize operationally meaningful quantum gains in a paradigmatic model of lossless multiple-phase interferometry and stress insufficiency of the analysis based solely on the concept of quantum Fisher information (QFI). We show that the advantage of the optimal simultaneous estimation scheme amounts to a constant factor improvement when compared with schemes where each phase is estimated separately -- contrary to the results obtained by a naive application of QFI, which leads to a better precision scaling in terms of the number of phases involved.
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