The existence of correlations between the parts of a quantum system on the one hand, and entanglement between them on the other, are different properties. Yet, one intuitively would identify strong $N$-party correlations with $N$-party entanglement in an $N$-partite quantum state. If the local systems are qubits, this intuition is confirmed: The state with the strongest $N$-party correlations is the Greenberger-Horne-Zeilinger (GHZ) state, which does have genuine multipartite entanglement. However, for high-dimensional local systems the state with strongest $N$-party correlations may be a tensor product of Bell states, that is, partially separable. We show this by introducing several novel tools for handling the Bloch representation.
 U. Fano, A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics, Phys. Rev. 93, 121 (1954).
 U. Fano, Description of States in Quantum Mechanics by Density Matrix and Operator Techniques, Rev. Mod. Phys. 29, 74 (1957).
 G. Mahler and V.A. Weberruß, Quantum Networks, 2nd Edition (Springer, Berlin, 2004).
 C. Klöckl and M. Huber, Characterizing multipartite entanglement without shared reference frames, Phys. Rev. A 91, 042339 (2015).
 C. Eltschka and J. Siewert, Monogamy equalities for qubit entanglement from Lorentz invariance, Phys. Rev. Lett. 114, 140402 (2015).
 M.-C. Tran, B. Dakic, F. Arnault, W. Laskowski, and T. Paterek, Quantum entanglement from random measurements, Phys. Rev. A 94, 042302 (2016).
 P. Appel, M. Huber, and C. Klöckl, Monogamy of correlations and entropy inequalities in the Bloch picture, J. Phys. Commun. (2020), doi:10.1088/2399-6528/ab6fb.
 F. Huber, O. Gühne, and J. Siewert, Absolutely Maximally Entangled States of Seven Qubits Do Not Exist, Phys. Rev. Lett. 118, 200502 (2017).
 C. Eltschka and J. Siewert, Distribution of entanglement and correlations in all finite dimensions, Quantum 2, 64 (2018).
 N. Wyderka, F. Huber, and O. Gühne, Constraints on correlations in multiqubit systems, Phys. Rev. A 97, 060101 (2018).
 F. Huber, C. Eltschka, J. Siewert, and O. Gühne, Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity, J. Phys. A: Math. Theor. 51, 175301 (2018).
 C. Eltschka, F. Huber, O. Gühne, and J. Siewert, Exponentially many entanglement and correlation constraints for multipartite quantum states Phys. Rev. A 98, 052317 (2018).
 T. Cox and P.C.E. Stamp, Partitioned density matrices and entanglement correlators, Phys. Rev. A 98, 062110 (2018).
 C. Eltschka and J. Siewert, Joint Schmidt-type decomposition for two bipartite pure states, Phys. Rev. A 101, 022302 (2020).
 J. Schlienz and G. Mahler, Description of entanglement, Phys. Rev. A 52, 4396 (1995).
 J. Schlienz and G. Mahler, The maximal entangled three-particle state is unique, Phys. Lett. A 224, 39 (1996).
 M. Żukowski and C. Brukner, Bell's theorem for general $N$-qubit states, Phys. Rev. Lett. 88, 210401 (2002).
 M. Teodorescu-Frumosu and G. Jaeger, Quantum Lorentz-group invariants of $n$-qubit systems, Phys. Rev. A 67, 052305 (2003).
 H. Aschauer, J. Calsamiglia, M. Hein, and H.J. Briegel, Local invariants for multi-partite entangled states allowing for a simple entanglement criterion, Quantum Inf. Comput. 4, 383 (2004); journal link; arXiv.org link.
 A. J. Scott, Multipartite entanglement, quantum error correcting codes, and entangling power of quantum evolutions, Phys. Rev. A 69, 052330 (2004).
 J.I. de Vicente, Further results on entanglement detection and quantification from the correlation matrix criterion, J. Phys. A: Math. Theor. 41, 065309 (2008).
 P. Badziag, C. Brukner, W. Laskowski, T. Paterek, and M. Żukowski, Experimentally Friendly Geometrical Criteria for Entanglement, Phys. Rev. Lett. 100, 140403 (2008).
 W. Laskowski, M. Markiewicz, T. Paterek, and M. Żukowski, Correlation-tensor criteria for genuine multiqubit entanglement, Phys. Rev. A 84, 062305 (2011).
 J.I. de Vicente and M. Huber, Multipartite entanglement detection from correlation tensors, Phys. Rev. A 84, 062306 (2011).
 We will use the term ``$k$-sector length'' instead of ``squared $k$-sector length'' following Ref. Tran2016. In the present context this does not lead to confusion.
 One may imagine very different correlation quantifiers, e.g., D. Girolami, T. Tufarelli, and C.E. Susa, Quantifying Genuine Multipartite Correlations and their Pattern Complexity, Phys. Rev. Lett. 119, 140505 (2017).
 J. Kaszlikowski, A. Sen De, U. Sen, V. Vedral, A. Winter, Quantum Correlation Without Classical Correlations, Phys. Rev. Lett. 101, 070502 (2008).
 C. Schwemmer, L. Knips, M.C. Tran, A. de Rosier, W. Laskowski, T. Paterek, and H. Weinfurter, Genuine Multipartite Entanglement without Multipartite Correlations, Phys. Rev. Lett. 114, 180501 (2015).
 M.C. Tran, M. Zuppardo, A. de Rosier, L. Knips, W. Laskowski, T. Paterek, and H. Weinfurter, Genuine $N$-partite entanglement without $N$-partite correlation functions, Phys. Rev. A 95, 062331 (2017).
 W. Klobus, W. Laskowski, T. Paterek, M. Wiesniak, and H. Weinfurter, Higher dimensional entanglement without correlations, Eur. Phys. J. D 73, 29 (2019).
 This relation corresponds to a special case of the quantum MacWilliams identity, cf. Ref. Huber2018.
 V. Coffman, J. Kundu, and W.K. Wootters, Distributed entanglement, Phys. Rev. A 61, 052306 (2000).
 P. Rungta, V. Buzek, C.M. Caves, M. Hillery, and G.J. Milburn, Universal state inversion and concurrence in arbitrary dimensions, Phys. Rev. A 64, 042315 (2001).
 M. Lewenstein, R. Augusiak, D. Chruściński, S. Rana, and J. Samsonowicz, Sufficient separability criteria and linear maps, Phys. Rev. A 93, 042335 (2016).
 An in-depth analysis of this projection operator will be carried out in forthcoming work.
 D. Goyeneche and K. Życzkowski, Genuinely multipartite entangled states and orthogonal arrays, Phys. Rev. A 90, 022316 (2014).
 D. Goyeneche, D. Alsina, J.I. Latorre, A. Riera, and K. Życzkowski, Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices, Phys. Rev. A 92, 032316 (2015).
 Andreas Ketterer, Nikolai Wyderka, and Otfried Gühne, "Entanglement characterization using quantum designs", Quantum 4, 325 (2020).
 Haiqing Huang, Irfan Ahmed, Ahmed Ali, Xin-wei Zha, Raymond Hon-Fu Chan, and Yanpeng Zhang, "Relations between the average bipartite entanglement and N-partite correlation functions", Laser Physics 32 7, 075201 (2022).
 Andreas Ketterer, Satoya Imai, Nikolai Wyderka, and Otfried Gühne, "Statistically significant tests of multiparticle quantum correlations based on randomized measurements", Physical Review A 106 1, L010402 (2022).
 Matthias Miller and Daniel Miller, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 378 (2021) ISBN:978-1-6654-1691-7.
 Cornelia Spee, "Certifying the purity of quantum states with temporal correlations", Physical Review A 102 1, 012420 (2020).
 Jens Siewert, "On orthogonal bases in the Hilbert-Schmidt space of matrices", Journal of Physics Communications 6 5, 055014 (2022).
 Vaishali Gulati, Arvind, and Kavita Dorai, "Classification and measurement of multipartite entanglement by reconstruction of correlation tensors on an NMR quantum processor", The European Physical Journal D 76 10, 194 (2022).
 Satoya Imai, Nikolai Wyderka, Andreas Ketterer, and Otfried Gühne, "Bound Entanglement from Randomized Measurements", Physical Review Letters 126 15, 150501 (2021).
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