# Techniques: Non-Destructive Zero-Knowledge Proofs on Quantum States, and Multi-Party Generation of Authorized Hidden GHZ States

## What the paper contributes (one paragraph)

The paper gives two orthogonal cryptographic capabilities and combines them into multi-party protocols. First, it defines and constructs Non-Interactive and Non-Destructive Zero-Knowledge Proofs on Quantum States (NIZKoQS): a sender transmits classical instructions for producing a quantum state together with a classical post-quantum NIZK proof about those instructions, so a receiver can certify non-trivial properties of the received state without destroying it, without interaction, and over a purely classical channel. Second, it improves classical-client Remote State Preparation (RSP), which fakes a quantum channel over a classical one, by preparing an n-qubit state (a GHZ state hidden among computational-basis qubits) from a single superposition, giving an asymptotic quadratic cost reduction, and lifts this to a multi-party setting. Combining the two yields a protocol for blind distribution of an authorized hidden GHZ state, where only applicants knowing a secret credential are entangled, with applications to quantum secret sharing, anonymous transmission, and onion routing. Security rests on Learning With Errors.

## Techniques offered

- **Non-interactive zero-knowledge proof on quantum states (NIZKoQS)** - certifies non-trivial properties of a received quantum state non-destructively, non-interactively, and over a classical channel by attaching a classical NIZK proof to the classical instructions that generate the state.
  - guarantee: interactive (zero-knowledge) soundness and zero-knowledge against a malicious verifier, computational under LWE; verification is non-destructive and reveals nothing beyond membership in the quantum language.
  - quote: "We initiate the study of \emph{Non-Interactive and Non-Destructive Zero-Knowledge Proofs on Quantum States} (NIZKoQS). We obtain these guarantees by sending a classical NIZK proof on \emph{classical} instructions used to produce a quantum state."

- **Single-superposition multi-qubit classical-client remote state preparation** - prepares an n-qubit hidden GHZ state on a quantum server from one superposition, faking a quantum channel over a classical channel, and extends to a multi-party authorized GHZ distribution.
  - guarantee: asymptotic quadratic efficiency gain, from O(nMN) to O((M+n)N), plus blindness of the support set and protection against malicious or noisy applicants altering the state, computational under LWE.
  - quote: "We first provide a method to classically prepare large states on $n$ qubits (where a \GHZ{} state is hidden in between $\ket{0}$ and $\ket{1}$ qubits) using a single superposition."

## Where it could apply

- Target primitive(s): remote state preparation, GHZ and Bell state distribution, state certification (non-destructive property test); not VQE/QAOA/QPE/Trotter sampling.
- Target application group(s): crypto_security, networking (quantum anonymous transmission, quantum secret sharing, quantum onion routing), platforms_tooling.

## Caveats

Security is computational, resting on the hardness of Learning With Errors with superpolynomial modulus-to-noise ratio; an unbounded verifier could learn the full produced state, so the guarantees are not statistical. The LWE instantiation of the cryptographic function family is a building block internal to the RSP and NIZKoQS constructions rather than a separately suppliable capability, and the impossibility result for the desirable blindCan variant is a no-go bound, not a supplied technique. The constructions are theoretical protocols with asymptotic costs (large constants M greater than N), not benchmarked implementations.