One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.
 Teiko Heinosaari, Takayuki Miyadera, and Mário Ziman. An invitation to quantum incompatibility. J. Phys. A, 49:123001, 2016.
 Marco Túlio Quintino, Tamás Vértesi, and Nicolas Brunner. Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality. Phys. Rev. Lett., 113:160402, 2014.
 Roope Uola, Tobias Moroder, and Otfried Gühne. Joint measurability of generalized measurements implies classicality. Phys. Rev. Lett., 113:160403, 2014.
 Roope Uola, Costantino Budroni, Otfried Gühne, and Juha-Pekka Pellonpää. One-to-one mapping between steering and joint measurability problems. Phys. Rev. Lett., 115:230402, 2015.
 Jukka Kiukas, Constantino Budroni, Roope Uola, and Juha-Pekka Pellonpää. Continuous-variable steering and incompatibility via state-channel dualism. Phys. Rev. A, 96:042331, 2017.
 Armin Tavakoli and Roope Uola. Measurement incompatibility and steering are necessary and sufficient for operational contextuality. Phys. Rev. Research, 2:013011, 2020.
 Lucas Clemente and Johannes Kofler. Necessary and sufficient conditions for macroscopic realism from quantum mechanics. Phys. Rev. A, 91:062103, 2015.
 Roope Uola, Giuseppe Vitagliano, and Costantino Budroni. Leggett-Garg macrorealism and the quantum nondisturbance conditions. Phys. Rev. A, 100:042117, 2019.
 Claudio Carmeli, Teiko Heinosaari, and Alessandro Toigo. Quantum incompatibility witnesses. Phys. Rev. Lett., 122:130402, 2019.
 Paul Skrzypczyk, Ivan Šupić, and Daniel Cavalcanti. All sets of incompatible measurements give an advantage in quantum state discrimination. Phys. Rev. Lett., 122:130403, 2019.
 Roope Uola, Tristan Kraft, Jiangwei Shang, Xiao-Dong Yu, and Otfried Gühne. Quantifying quantum resources with conic programming. Phys. Rev. Lett., 122:130404, 2019.
 Leonardo Guerini, Marco Túlio Quintino, and Leandro Aolita. Distributed sampling, quantum communication witnesses, and measurement incompatibility. Phys. Rev. A, 100:042308, 2019.
 Francesco Buscemi, Eric Chitambar, and Wenbin Zhou. Complete resource theory of quantum incompatibility as quantum programmability. Phys. Rev. Lett., 124:120401, 2020.
 Michael M. Wolf, David Perez-Garcia, and Carlos Fernandez. Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory. Phys. Rev. Lett., 103:230402, 2009.
 National Research Council. Mathematical Challenges from Theoretical/Computational Chemistry. The National Academies Press, Washington, DC, 1995.
 Yi-Kai Liu. Consistency of local density matrices is QMA-complete. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg, 2006.
 Christian Schilling, Carlos L. Benavides-Riveros, and Péter Vrana. Reconstructing quantum states from single-party information. Phys. Rev. A, 96:052312, 2017.
 Giulio Chiribella. On quantum estimation, quantum cloning and finite quantum de Finetti theorems. In Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science, vol 6519. Springer, Berlin, Heidelberg, 2011.
 Teiko Heinosaari and Takayuki Miyadera. Incompatibility of quantum channels. J. Phys. A, 50:135302, 2017.
 Eneet Kaur, Siddhartha Das, Mark M. Wilde, and Andreas Winter. Extendibility limits the performance of quantum processors. Phys. Rev. Lett., 123:070502, 2019.
 Andrew C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Distinguishing separable and entangled states. Phys. Rev. Lett., 88:187904, 2002.
 Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A, 223:1, 1996.
 Kai Sun, Xiang-Jun Ye, Ya Xiao, Xiao-Ye Xu, Yu-Chun Wu, Jin-Shi Xu, Jing-Ling Chen, Chuan-Feng Li, and Guang-Can Guo. Demonstration of Einstein–Podolsky–Rosen steering with enhanced subchannel discrimination. npj Quantum Information, 4:12, 2018.
 Wenqiang Zheng, Zhihao Ma, Hengyan Wang, Shao-Ming Fei, and Xinhua Peng. Experimental demonstration of observability and operability of robustness of coherence. Phys. Rev. Lett., 120:230504, 2018.
 Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge University Press, Cambridge, 2004.
 Bernd Gärtner and Jiří Matoušek. Approximation Algorithms and Semidefinite Programming. Springer Verlag, Berlin, Heidelberg, 2012.
 J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White. Ancilla-assisted quantum process tomography. Phys. Rev. Lett., 90:193601, 2003.
 Ludovico Lami, Sumeet Khatri, Gerardo Adesso, and Mark M. Wilde. Extendibility of bosonic gaussian states. Phys. Rev. Lett., 123:050501, 2019.
 Ryuji Takagi, Bartosz Regula, Kaifeng Bu, Zi-Wen Liu, and Gerardo Adesso. Operational advantage of quantum resources in subchannel discrimination. Phys. Rev. Lett., 122:140402, 2019.
 Ryuji Takagi and Bartosz Regula. General resource theories in quantum mechanics and beyond: operational characterization via discrimination tasks. Phys. Rev. X, 9:031053, 2019.
 C.W. Gardiner and P. Zoller. Quantum Noise, A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd Edition, Springer Verlag, Berlin Heidelberg, 2004.
 Martin Plávala, "General probabilistic theories: An introduction", Physics Reports 1033, 1 (2023).
 Chung-Yun Hsieh, Matteo Lostaglio, and Antonio Acín, "Quantum channel marginal problem", Physical Review Research 4 1, 013249 (2022).
 Otfried Gühne, Erkka Haapasalo, Tristan Kraft, Juha-Pekka Pellonpää, and Roope Uola, "Colloquium : Incompatible measurements in quantum information science", Reviews of Modern Physics 95 1, 011003 (2023).
 Cristhiano Duarte, Lorenzo Catani, and Raphael C. Drumond, "Relating Compatibility and Divisibility of Quantum Channels", International Journal of Theoretical Physics 61 7, 189 (2022).
 Xian Shi, "Entanglement polygon inequalities for pure states in qudit systems", The European Physical Journal Plus 138 8, 768 (2023).
 Mark Girard, Martin Plávala, and Jamie Sikora, "Jordan products of quantum channels and their compatibility", Nature Communications 12, 2129 (2021).
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