Where the P vs NP question stands: circuit and proof-complexity lower bounds, barriers, meta-complexity, geometric complexity theory, new reductions and hardness results, and how the community reads each claimed proof.
A new arXiv paper proves a quantum version of the Razborov-Rudich natural proofs barrier. Under a standard cryptographic assumption, no natural property of quantum states can prove superpolynomial state-preparation lower bounds, even against a fixed level of the Magic Hierarchy. Existing techniques based on approximate degree, unique ground states of local Hamiltonians and mutual information are natural in this sense.
A new arXiv paper gives deterministic algorithms running in O(n^1.9992) time for 3SUM and O(n^2.9995) time for APSP, which would refute the 3SUM and APSP hypotheses. Known reductions would also refute the Exact Triangle, Zero-Weight k-Clique and several Online Matrix-Vector hypotheses, removing core assumptions of fine-grained complexity.
Also in cellcog.ai, mindpattern.ai, aiweekly.coAssuming the Sparse MAX-3-SAT hypothesis, evaluating k-variable queries between PP^k and FO^k in O(m^{k-ε}) time is tractable only for three-variable, two-variable and unary-signature fragments. For every k ≥ 4 and ε > 0, a fixed PP^k sentence over one binary relation cannot be evaluated in that time, a conditional fine-grained lower bound.
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Real progress on the six open Millennium Prize Problems — the Riemann hypothesis, P vs NP, Navier–Stokes, Yang–Mills, Hodge and Birch–Swinnerton-Dyer: serious papers, partial results, AI-assisted advances, and what experts say about claimed solutions.
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Progress in quantum hardware and error correction — experiments on real devices, not just proposals.