Multisetting Greenberger-Horne-Zeilinger theorem

We present a generalized Greenberger-Horne-Zeilinger (GHZ) theorem, which involves more than two local measurement settings for some parties, and cannot be reduced to one with less settings. Our results hold for an odd number of parties. We use a set of observables, which are incompatible but share a common eigenstate, here a GHZ state. Such observables are called concurrent. The idea is illustrated with an example of a three-qutrit system and then generalized to systems of higher dimensions, and more parties. The GHZ paradoxes can lead to, e.g., secret sharing protocols.We thank M. Wiesniak, T. Paterek, and J. Jae for comments. This work is supported by the Foundation for Polish Science TEAM project cofinanced by the EU European Regional Development Fund, and by a NCBiR-CHIST-ERA Project QUASAR. R. R. acknowledges support by an ERC grant QOLAPS (Grant No. 291348). C. L. and D. A. are supported by the National Research Foundation and Ministry of Education (partly through the Tier 3 Grant "Random numbers from quantum processes"), Singapore. J. L. is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (Grants No. 2010-0015059 and No. 2010-0018295)