Violations of multisetting quaternion and octonion Bell inequalities

We examine the N-partite quaternion and octonion dichotomic Bell inequalities derived by Vogel and Shchukin utilizing the square identities of Euler and Degen, which apply to experiments with M settings at each spatially separated site. We reveal these inequalities to be violated by the Greenberger-Horne-Zeilinger state, for N≥3 and 2≤M≤8. Violations are robust with respect to loss, the threshold detection efficiency being η>22N-1 for all M, implying violations for efficiencies as low as η∼50% at each site, as N→.