We show that one-way one-round quantum-LOCAL algorithms cannot $4$-color directed cycles with high probability. This is the first lower bound in the high-probability quantum LOCAL setting that goes beyond the non-signaling and bounded-dependence models, exploiting the structure of distributed quantum algorithms. Our proof establishes a bidirectional connection between distributed quantum computing and extremal combinatorics. We obtain our lower bound by proving a Mantel-type stability theorem for weighted matrix spaces.
We prove a nearly optimal lower bound on the blocklength of general relaxed locally decodable codes (RLDCs), extending the result of [GGS26] to nonlinear encodings and adaptive decoders with two-sided error. Namely, we show that any $q$-query RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+Ω(1/q)}$, closely matching the upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof reduces showing RLDC lower bounds to estimating the output distribution of adaptive decoders. To this end, we introduce a capped Horvitz--Thompson estimator with uniformly bounded discrete derivatives on the $p$-biased cube. We establish its concentration of measure via self-bounding inequalities and control the downward bias induced by capping through a spreadness argument.