The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
 J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics 1, 195 (1964).
 C. Branciard, N. Gisin, and S. Pironio. Characterizing the Nonlocal Correlations Created via Entanglement Swapping. Phys. Rev. Lett. 104, 170401 (2010).
 T. Fritz. Beyond Bell's theorem: correlation scenarios. New J. Phys. 14, 103001 (2012).
 J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens. Quantum common causes and quantum causal models. Phys. Rev. X 7, 031021 (2017).
 F. Costa and S. Shrapnel. Quantum causal modelling. New Journal of Physics 18, 063032 (2016).
 D. Gross, M. Müller, R. Colbeck, and O. C. O. Dahlsten. All reversible dynamics in maximally nonlocal theories are trivial. Phys. Rev. Lett. 104, 080402 (2010).
 J. Pearl. Causality, (Cambridge University Press 2009).
 J. Henson, R. Lal, and M. F. Pusey. Theory-independent limits on correlations from generalized Bayesian networks. New J. Phys. 16, 113043 (2014).
 J. Pearl. On the Testability of Causal Models with Latent and Instrumental Variables. In Proc. 11th Conf. Uncertainty in Artificial Intelligence, pages 435-443 ( 1995).
 B. Bonet. Instrumentality Tests Revisited. In Proc. 17th Conf. Uncertainty in Artificial Intelligence, pages 48-55 ( 2001).
 R. Chaves, G. Carvacho, I. Agresti, V. D. Giulio, L. Aolita, S. Giacomini, and F. Sciarrino. Quantum violation of an instrumental test. Nat. Phys. 47, 291–296 (2018).
 J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett. 23, 880 (1969).
 E. Wolfe et al. Causal Inference for Generalized Bayesian Networks. In preparation.
 R. Chaves, R. Kueng, J. B. Brask, and D. Gross. Unifying Framework for Relaxations of the Causal Assumptions in Bell's Theorem. Phys. Rev. Lett. 114, 140403 (2015).
 J. Barrett and N. Gisin. How Much Measurement Independence Is Needed to Demonstrate Nonlocality? Phys. Rev. Lett. 106, 100406 (2011).
 G. Pütz, D. Rosset, T. J. Barnea, Y.-C. Liang, and N. Gisin. Arbitrarily Small Amount of Measurement Independence is Sufficient to Manifest Quantum Nonlocality. Phys. Rev. Lett. 113, 190402 (2014).
 M. Navascués, S. Pironio, and A. Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys. 10, 073013 (2008).
 M. Navascués, G. de la Torre, and T. Vértesi. Characterization of Quantum Correlations with Local Dimension Constraints and Its Device-Independent Applications. Phys. Rev. X 4, 011011 (2014).
 M. Navascués and T. Vértesi. Bounding the Set of Finite Dimensional Quantum Correlations. Phys. Rev. Lett. 115, 020501 (2015).
 J. Sikora, A. Varvitsiotis, and Z. Wei. Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation. Phys. Rev. Lett. 117, 060401 (2016).
 C. Jones, E. C. Kerrigan, and J. Maciejowski. Equality Set Projection: A new algorithm for the projection of polytopes in halfspace representation. Technical report, Cambridge University Engineering Dept (2004).
 S. I. Bastrakov and N. Y. Zolotykh. Fast method for verifying Chernikov rules in Fourier-Motzkin elimination. Comp. Mat. & Math. Phys. 55, 160 (2015). ISSN 1555-6662.
 T. Christof and A. Löbel. PORTA - POlyhedron Representation Transformation Algorithm (2009).
 S. Pironio. All Clauser-Horne-Shimony-Holt polytopes. J. Phys. A 47, 424020 (2014). ISSN 1751-8121.
 A. Acín, S. Massar, and S. Pironio. Randomness versus nonlocality and entanglement. Phys. Rev. Lett. 108, 100402 (2012).
 L. K. Shalm et al. Strong Loophole-Free Test of Local Realism. Phys. Rev. Lett. 115, 250402 (2015).
 M. Giustina et al. Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons. Phys. Rev. Lett. 115, 250401 (2015).
 D. Mayers and A. Yao. Quantum cryptography with imperfect apparatus. In Proc. 39th Symposium on Foundations of Computer Science, pages 503-509 ( 1998).
 A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani. Device-Independent Security of Quantum Cryptography against Collective Attacks. Phys. Rev. Lett. 98, 230501 (2007).
 R. Colbeck and A. Kent. Private randomness expansion with untrusted devices. J. Phys. A 44, 095305 (2011). ISSN 1751-8121. Earlier version published in R. Colbeck, ``Quantum and relativistic protocols for secure multi-party computation", PhD thesis, Cambridge, 2006.
 C. A. Miller and Y. Shi. Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices. In Proc. 46th Symposium on Theory of Computing, page 417 ( 2014).
 C. Branciard, D. Rosset, N. Gisin, and S. Pironio. Bilocal versus nonbilocal correlations in entanglement-swapping experiments. Phys. Rev. A 85, 032119 (2012).
 Elie Wolfe, David Schmid, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes", arXiv:1903.06311.
 Iris Agresti, Davide Poderini, Leonardo Guerini, Michele Mancusi, Gonzalo Carvacho, Leandro Aolita, Daniel Cavalcanti, Rafael Chaves, and Fabio Sciarrino, "Experimental device-independent certified randomness generation with an instrumental causal structure", arXiv:1905.02027.
 Giulio Chiribella and Daniel Ebler, "Quantum speedup in the identification of cause-effect relations", Nature Communications 10, 1472 (2019).
 Miguel Navascues and Elie Wolfe, "The Inflation Technique Completely Solves the Causal Compatibility Problem", arXiv:1707.06476.
 Debasis Mondal, Jaskaran Singh, and Dagomir Kaszlikowski, "No nonlocal advantage of quantum coherence beyond quantum instrumentality", arXiv:1901.07008.
 Alejandro Pozas-Kerstjens, Rafael Rabelo, Łukasz Rudnicki, Rafael Chaves, Daniel Cavalcanti, Miguel Navascués, and Antonio Acín, "Bounding the Sets of Classical and Quantum Correlations in Networks", Physical Review Letters 123 14, 140503 (2019).
 Mirjam Weilenmann and Roger Colbeck, "Analysing causal structures in generalised probabilistic theories", arXiv:1812.04327.
 Ana Belén Sainz, Matty J. Hoban, Paul Skrzypczyk, and Leandro Aolita, "Bipartite post-quantum steering in generalised scenarios", arXiv:1907.03705.
 G. Carvacho, R. Chaves, and F. Sciarrino, "Perspective on experimental quantum causality", EPL (Europhysics Letters) 125 3, 30001 (2019).
 Andrew J. P. Garner, Marius Krumm, and Markus P. Mueller, "Semi-device-independent information processing with spatiotemporal degrees of freedom", arXiv:1907.09274.
 Davide Poderini, Rafael Chaves, Iris Agresti, Gonzalo Carvacho, and Fabio Sciarrino, "Exclusivity graph approach to Instrumental inequalities", arXiv:1909.09120.
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