The problem of quantum dice rolling (DR)-a generalization of the problem of quantum coin flipping (CF) to more than two outcomes and partiesis studied in both its weak and strong variants. We prove by construction that quantum mechanics allows for (i) weak N -sided DR admitting arbitrarily small bias for any N and (ii) two-party strong N -sided DR saturating Kitaev's bound for any N . To derive (ii) we also prove by construction that quantum mechanics allows for (iii) strong imbalanced CF saturating Kitaev's bound for any degree of imbalance. Furthermore, as a corollary of (ii) we introduce a family of optimal 2m-party strong n m -sided DR protocols for any pair m and n.

BYThis formula is also applicable to a non-parabolic band with plane waves as wave functions because it does not contain the energy Q and the effective mass m* explicitly. Taking into account the k-dependence of the Bloch amplitudes *) All the formulae which appear further below involve the effective maas near the band edge, thus the subscript "0" of the effective mass mz, m&, and rnz is omitted.

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