Reading and Working with Active Conjectures

A conjecture is a proposition whose status depends on an exact formulation. A research program can contain many inequivalent conjectures with similar names. Before choosing one, specify its variables, quantifiers, model class, approximation, and intended limit, then verify the current status of that formulation.

Write the quantifiers before the motivation

“Generic systems thermalize” is too ambiguous for a proof or counterexample. Which Hamiltonians, initial states, observables, and times are included? Does the claim concern almost all eigenstates or every eigenstate? Is the limit taken in system size before time? A proposed counterexample may violate an unstated qualifier rather than the intended proposition.

The stable Mathematical Quantum Mechanics volume supplies the language for domains, spectra, and limits. Research assessments such as ETH and many-body localization identify which physical distinctions make a formulation consequential.

Hamiltonian complexity provides a useful example. A result about low-energy states, a local-Hamiltonian approximation problem, and a nonlocal-game formulation can address related ideas without proving the same conjecture. The quantum complexity assessment separates these statements and includes a 2024 status correction for the games literature.

Anshu, Breuckmann, and Nirkhe’s NLTS construction is a specific advance. It should not be cited as if its title established every formulation of the quantum PCP conjecture. Natarajan and Nirkhe’s status note illustrates why source versions and corrections matter. This page uses those verified sources as a methodological example, not as an exhaustive October 2026 status census.

Three accessible forms of contribution

Restricted theorem. Prove the statement for a class in which one difficult feature is controlled: dimension, commutativity, locality, symmetry, or a spectral assumption. Explain why the restriction is meaningful and where the proof uses it.

Counterexample or obstruction. Construct a valid example that violates a precisely stated extension. A small exact example can be decisive against a universal claim. A numerical search that finds none is not a proof.

Discriminating diagnostic. Identify a measurable or computable quantity on which plausible scenarios disagree. State the finite-size, finite-time, and numerical limitations that prevent it from becoming a theorem about an asymptotic limit.

Starter exercise: audit one claimed implication

Choose a short lemma or reduction from a cited source. List every input and output parameter, reconstruct one finite example, and trace the hypothesis used at each step. Then remove one assumption and seek a minimal failure.

The deliverable is a two-page proof audit or counterexample, with exact conventions and a reference to the original statement. Use symbolic or numerical computation to test examples, but keep the argument distinct from empirical search. The exercise becomes research when it yields a new corrected statement, a stronger bound, or a gap that the literature has not already closed.

Decide when to reassess

A new theorem changes the assessment only if its statement matches the proposed problem. A proof correction can reopen a step without invalidating all related work. Read the technical claim, not only the abstract’s broad description, and seek expert feedback before investing in a famous conjecture as a first project.

References

A. Anshu, N. P. Breuckmann, and C. Nirkhe, “NLTS Hamiltonians from good quantum codes,” Proceedings of STOC 2023, 1090–1096 (2023). Preprint.
A. Natarajan and C. Nirkhe, “The status of the quantum PCP conjecture (games version),” arXiv:2403.13084 (2024). Status note.
P. Sierant et al., “Many-body localization in the age of classical computing,” Reports on Progress in Physics 88, 026502 (2025). Preprint.