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Can entangling codewords across independent uses of a quantum channel improve the rate of classical communication? An old conjecture states that there is no such entanglement advantage for all channels. Although Hastings's result and follow-up works disproved this conjecture, it turns out that enormous dimension is required to construct a channel where entanglement enhances classical communication. This indicates that there might be no entanglement advantage for many physical channels, or for all low dimensional channels. Motivated by recent progress on (generalized) amplitude damping channels, we propose a general Schur-weighted block entropy inequality, and show that there is no entanglement advantage for all phase-covariant channels including multi-level amplitude damping channels, generalized dephasing channels, and damping-dephasing channels. As a by-product, we also recover the recently established result that there is no entanglement advantage for all qubit channels.

Advanced power adjusting block withholding attacks such as PAW and T-PAW have been analyzed so far under a global-reset version of the proportional scheme (GRP), which is rarely used in practice. In this paper, we analyze T-PAW, which contains FAW and PAW as special cases, under the block-dependent payout schemes deployed in practice: Pay-Per-Last-N-Shares (PPLNS), round-based Proportional (PROP) and exponential-decay (EXP). We derive the revenue ratios analytically, by renewal arguments for PPLNS and through the stationary carry process of an open round for PROP and EXP. We further introduce Power Adjusting Hopping (PAH), where an adversary of any size adjusts its power allocation towards the pool according to the blockwork age of the round without withholding any block, and derive its revenue ratios under PROP and EXP. We show that GRP misestimates the T-PAW gain in a scheme-dependent direction: it overstates the gain under PROP up to an order of magnitude and understates it under EXP for the original slush decay rate. PROP is highly vulnerable to PAH, while EXP faces a trade-off: a slow decay invites PAH and a fast decay invites T-PAW. PPLNS is the least vulnerable scheme: it is immune to PAH and it mitigates, but does not eliminate, T-PAW. All analytical results are verified by Monte Carlo simulations.

Predicting how many tokens a large language model (LLM) will decode, using only prefill-phase hidden states, is a key primitive for inference scheduling. We show that current benchmarks conflate predictor quality with task hardness: when the target model, dataset, or sampling configuration changes, reported accuracy collapses even though the predictor is unchanged. We trace this instability to an overlooked error source---the inherent stochasticity of autoregressive decoding---and derive bounds that every prefill-side method must obey. Our analysis treats decoding as a stopped stochastic process. The per-step stopping probability of a response is not constant but ramps up over hundreds of steps, so the decode length inherits a central limit theorem from the many local decisions that determine where a response ends; this yields per-prompt normality with an explicit error term, together with an affine variance law $σ=αμ+b$ whose slope is the coefficient of variation of the number of response segments. Empirically, the normal law reproduces the per-prompt MAE and the central mass of the decode-length distribution to within a few percent, and the residual departures from normality (mild right skew, slight excess kurtosis) are the ones predicted by the sampling-noise correction. Under fixed model, data, and sampling conditions, a broad design space of predictors (MLP, EGTP, FiLM, soft-label, and Mixture-of-Experts variants) converges to similar performance close to the bound, confirming that \emph{system error dominates model error}. We argue for a benchmarking protocol that fixes the model, data, and sampling parameters and that reports relative-error metrics alongside absolute ones. Code and analysis tools are available at https://github.com/ShwStone/DecodingPredict.

We develop design methodologies for the rate profile of polarization-adjusted convolutional (PAC) codes tailored to successive cancellation list (SCL) decoding. The crux of our approach is two-fold: first is the creation of a modified density evolution Gaussian approximation (DEGA), called conditional DEGA (CDEGA), to analyze the decoding process in the presence of errors; second is the creation of a dangerous-path occupancy metric to find the conditional occupancy of bit-errors in a finite list. Our approach not only illuminates the role of rate-one precoding, but also provides insight into why certain information sets are particularly well suited to PAC codes. We demonstrate the usefulness of CDEGA and dangerous-path metrics with algorithms that create new information sets, as well as precoders that are extremely sparse yet competitive. In particular, we design codes of length $N \leq 512$ with various rates and list sizes, all of which offer at least half a dB gain over traditional PAC codes at an FER of $10^{-3}$ and improve upon state-of-the-art results.

Quantum error correction is essential for fault-tolerant quantum computing. High-rate quantum low-density parity-check (qLDPC) codes have emerged as a promising approach to reducing the substantial physical qubit overhead of quantum error correction, enabling encoding rates exceeding $1/2$ in blocks of approximately one thousand physical data qubits. However, their practical implementation requires decoders that combine high accuracy with low latency to keep pace with quantum hardware. Existing decoders of high-rate qLDPC codes can achieve high accuracy but often incur substantial computational latency, whereas neural network decoders have yet to be demonstrated for high-rate qLDPC codes. Here, we introduce \emph{neural triage decoding}, a framework that uses syndrome consistency to adaptively allocate neural computation. A local neural message-passing network rapidly predicts error corrections, additional iterations reduce residual syndrome violations, and a rarely invoked diversification stage searches for consistent corrections to the remaining shots. We demonstrate that the \texttt{Triage} decoder can decode ultra-high-rate qLDPC codes encoding a few hundred logical qubits, including lifted-product and Kasai codes, achieving state-of-the-art accuracy comparable to the near optimal algorithmic decoders while reducing decoding latency by approximately one to two orders of magnitude. Our framework is inspired by a rigorous analysis of the exact bitwise marginal decoder, revealing a distance-independent syndrome-inconsistent error floor. In addition, we show the robustness of neural network decoders to mismatched error priors. Our results establish adaptive neural decoding as a promising route towards fast, accurate, and scalable quantum error correction in the near term.

In this note, we prove that for a subset $\mathcal{F}$ of $\{0,1\}^n$ of density $\frac{1}{8}$, its average pairwise Hamming distance is at least $\frac{n-3}{2}$, with equality attained only by $(n-3)$-dimensional subcubes. This answers a question of Lei Yu, and solves an important open case of a more general question posed by Ahlswede and Katona in 1977.

We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_φ(ρ) = \operatorname{tr}(φ(dρ))/d$, where $ρ$ is an unknown $d$-dimensional quantum state and $φ$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_φ(ρ)$ from properties of $φ$, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is \[ Θ\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right), \] and the sample complexity of spectrum estimation is \[ Θ\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right). \] In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.