Existence of SIC-POVMs (Zauner's conjecture)
Prove that a symmetric informationally complete POVM (d^2 equiangular lines in C^d) exists in every dimension d, and extend the list of dimensions with exact or numerical solutions.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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Ground-state phase diagram of the doped 2D Hubbard model
Determine reliably, with controlled numerics, where the doped two-dimensional Hubbard model (with and without next-nearest-neighbour hopping t′) is superconducting, striped or otherwise ordered.
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Hadronic vacuum polarisation in the muon g−2
Resolve the disagreement between lattice-QCD and data-driven (e+e− → hadrons) evaluations of the leading hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.
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Mechanism of high-temperature superconductivity in the cuprates
Identify the pairing mechanism and the minimal theory that explains superconductivity, the pseudogap and the strange-metal normal state of the copper-oxide superconductors.
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Mutually unbiased bases in dimension 6
Decide whether four (or seven) mutually unbiased bases exist in C^6; only three are known, and a complete set of seven is widely believed not to exist.
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Small-scale problems of cold dark matter
Decide whether the cusp–core, too-big-to-fail and rotation-curve diversity problems of ΛCDM on galaxy scales are explained by baryonic physics, by modified dark-matter properties (e.g. self-interactions), or by observational systematics.
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The cosmological lithium problem
Explain why the lithium-7 abundance observed in old metal-poor halo stars is a factor of about 3–4 below the prediction of standard Big Bang nucleosynthesis with the CMB baryon density.
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The Hubble tension
Explain why local distance-ladder measurements of the Hubble constant H0 disagree with the value inferred from the cosmic microwave background under ΛCDM, or show that the disagreement is systematic.
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The solar coronal heating problem
Determine which physical mechanisms heat the solar corona to millions of kelvin above a photosphere of about 5,800 K, and in what proportion, using public spacecraft data and simulations.
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The strong CP problem
Explain why the CP-violating θ parameter of QCD is experimentally smaller than about 10^-10 when nothing in the Standard Model requires it to be small.
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Theory of the glass transition
Determine whether the dramatic slowdown of supercooled liquids reflects an underlying thermodynamic phase transition (such as an ideal glass at a Kauzmann temperature) or is purely dynamical.
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Open Quantum Problem 13: Mutually unbiased bases
Special case in dimension 6: determine the maximal number of mutually unbiased orthonormal bases in ℂ^6.
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Open Quantum Problem 23: SIC-POVMs
Benchmark open subproblem: existence of a SIC-POVM in dimension 56.
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Open Quantum Problem 35: existence of absolutely maximally entangled pure states
Open benchmark statement: does an AME(8,4) state exist?
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