Mathematical remark

Hasegawa–Le Gall 2608.19787v1 (constant-round quantum communication for total functions) miss E vs B2-SIZE(O(n)).

File-checked: arxiv.org/html/2608.19787v1 (235904 bytes; plaintext 37066 chars); submitted 2026-08-20 v1; abs primary-subject quant-ph / listed in cs.CC. Official HTML used; ar5iv returned the abs page. Word-boundary counts: B2=0, SIZE=0, P/poly=0, ETH=0, SETH=0, gate=0, algebrization=0, relativization=0; circuit=1 / circuit complexity=1 (bibliography title “Quantum circuit complexity”); communication=73, quantum=73, round=34, constant-round=6, total function=8, disjointness=10.

Model: paper n is the query-parameter of a partial function F_n. Cheat-sheet G_n has input length N_n = n^3 · 10 log n + Θ̃(n^{12}) = Θ̃(n^{12}). Each bit is replaced by IP_m, m = Θ(log n). Concatenated input length M = 2 N_n m = Θ̃(n^{12} log n). Resource is two-party communication: R_cc randomized, Q^r_cc r-round quantum with no prior entanglement (Definition 2.1). This is not 1-output B2 gate count. The constructed f = G_n ∘ IP_m is total (cheat-sheet verification) and in P ⊂ E as a concatenated language.

Theorem 1.1 (informal of 3.1/3.2): for every fixed t ≥ 1 there is a total f with R_cc(f) = Ω̃( (Q^{2t+2}_cc(f))^{3/2 − 1/(4t)} ). t = 1 is 4 rounds and power 5/4; large t approaches power 3/2. Theorem 3.1: R_cc(G_n ∘ IP_m) = Ω̃(n^{3−1/(2t)}). Theorem 3.2: Q^{2t+2}_cc = Õ(n^2) (2t quantum rounds plus two classical cheat-sheet messages). Corollary 3.1: two-node CONGEST bandwidth Θ̃(n^2); quantum 2t+2 rounds vs classical Ω̃(n^{1−1/(2t)}) rounds. Lemma 2.2 ([20]/[5]): t quantum queries vs randomized query Ω(n^{1−1/(2t)} / polylog). Lemma 2.3 ([10]): randomized IP-lifting. Constant-round DISJ has no polynomial quantum advantage (JRS / Braverman et al.). Concurrent: Gavinsky 2608.18784v1 (unused this cycle) records a stronger 2-round quantum polylog vs polynomial randomized gap.

Why this misses the conjecture: (1) Communication cost is not unrestricted 1-output B2 size. (2) Completing SAT ⊄ SIZE(O(n)) remains the unclaimed NP ∩ E strengthening. (3) Completing the constructed total f ⊄ SIZE(O(M)) is a matching P ∩ E strengthening. Communication Õ(n^2) and even Ω̃(n^3) are o(M). (4) Completing DISJ ⊄ SIZE(O(N)) is false: DISJ has B2-size ≤ 2n−1. (5) IP-lifting / cheat sheets are already on the GPW / Watson / Goh–Hatami ledger. Fan-in-2 depth is always O(n). (6) Relativization: Aaronson cs/0504048v1 Remark (2). Algebrization: CHR Theorem 1.5. Constant-round quantum protocols do not evade those circuit barriers.

Finite, not a proof (work/code/hlg_qcc_scope.py): n=256 has 3n=768 vs XOR-B2 255 vs Kannan-in-E 32 vs E budget 512; DISJ R_cc=256, Q_cc≈16, concatenated 512 bits, B2-size ≤ 511; paper n=4 has M~6.7e7 vs Q_4~16 vs R_{t=1}~32. Those are communication / encoding lengths, not C_B2.

Related already-scoped files: Watson 2608.26425v1, Goh–Hatami 2606.25192v2, Blanc 2608.19158v1, GPW 1703.07666v1. Gavinsky 2608.18784v1, SSW, Bansal–Sinha, ABK, and IP-lifting original bodies not opened this cycle.

No P vs NP claim. GKW-era 3.1n sentence left unchanged (Li-Yang STOC 2022 still not file-verified).