Topological phases of matter can be protected not only by on-site symmetries, but also by crystalline symmetries. In this talk, I will present the classification of 1D symmetry-protected topological (SPT) phases protected by both on-site and crystalline symmetries—referred to as modulated SPT phases—and show how it aligns with the crystalline equivalence principle through the MPS formalism. Furthermore, I will discuss their applications to Lieb–Schultz–Mattis (LSM) constraints and non-invertible symmetries.

Relativistic zero-knowledge proofs (RZKP) exploit the no-superluminal principle between spatially separated provers, offering a use case of phishing-proof identity verification — but prior demonstrations required 60 meters of separation, and soundness against entangled provers remained open. In this paper (https://arxiv.org/abs/2507.14324), we address both. From an engineering perspective, we push the entire challenge-response path onto FPGAs and account for latency transceiver-by-transceiver, cutting the round-trip window to 100 ns and halving the required separation to 30 meters. These techniques are protocol-agnostic and transfer to any relativistic protocol. Deploying across two rooms of a working office building, we scale to six million rounds and provide the first phase-by-phase timing decomposition of an RZKP implementation. The bottleneck is not the physics. It is network packet processing during verification — isolated by a single-verifier three-prover variant that eliminates inter-verifier communication. On the theory side, we prove the entangled-soundness bound for a two-prover graph-coloring RZKP, via a modified protocol and a reduction through almost-commuting operator assignments. The bound is real, but loose: |E|⁸ scaling, against the |E|⁴ of the known three-prover result. We identify exactly where each penalty is paid, offering the reduction chain as a reusable template for lifting classical soundness proofs — and a map of its costs.