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Quantum Merkle Trees

Lijie Chen, Ramis Movassagh

TL;DR

The paper addresses the challenge of creating a quantum analog of the Merkle tree for committing to quantum information by introducing the Quantum Haar Random Oracle Model (QHROM) and a quantum Merkle tree that uses a Haar-random unitary $ ait G$ and its inverse. It then proposes a succinct quantum argument for the Gap-$k$-Local-Hamiltonian problem within this model, drawing a parallel to Kilian’s succinct NP argument but under quantum assumptions and the QPHC conjecture, via a tree-based commitment scheme and a decommitment procedure that reveals queried qubits. Completeness and (partial) soundness results are established, with the protocol achieving polylogarithmic communication and polynomial-time verifier operation under certain conditions, and a broader, fully general security proof remaining an open challenge. The work highlights significant open questions, including the development of a quantum analogue to compressed oracle techniques for QHROM, extensions to zero-knowledge, and potential generalizations to all of $ extsf{QMA}$, as well as connections to subsequent commitment-security results in the literature.

Abstract

Committing to information is a central task in cryptography, where a party (typically called a prover) stores a piece of information (e.g., a bit string) with the promise of not changing it. This information can be accessed by another party (typically called the verifier), who can later learn the information and verify that it was not meddled with. Merkle trees are a well-known construction for doing so in a succinct manner, in which the verifier can learn any part of the information by receiving a short proof from the honest prover. Despite its significance in classical cryptography, there was no quantum analog of the Merkle tree. A direct generalization using the Quantum Random Oracle Model (QROM) does not seem to be secure. In this work, we propose the quantum Merkle tree. It is based on what we call the Quantum Haar Random Oracle Model (QHROM). In QHROM, both the prover and the verifier have access to a Haar random quantum oracle $G$ and its inverse. Using the quantum Merkle tree, we propose a succinct quantum argument for the Gap-$k$-Local-Hamiltonian problem. Assuming the Quantum PCP conjecture is true, this succinct argument extends to all of QMA. This work raises a number of interesting open research problems.

Quantum Merkle Trees

TL;DR

The paper addresses the challenge of creating a quantum analog of the Merkle tree for committing to quantum information by introducing the Quantum Haar Random Oracle Model (QHROM) and a quantum Merkle tree that uses a Haar-random unitary and its inverse. It then proposes a succinct quantum argument for the Gap--Local-Hamiltonian problem within this model, drawing a parallel to Kilian’s succinct NP argument but under quantum assumptions and the QPHC conjecture, via a tree-based commitment scheme and a decommitment procedure that reveals queried qubits. Completeness and (partial) soundness results are established, with the protocol achieving polylogarithmic communication and polynomial-time verifier operation under certain conditions, and a broader, fully general security proof remaining an open challenge. The work highlights significant open questions, including the development of a quantum analogue to compressed oracle techniques for QHROM, extensions to zero-knowledge, and potential generalizations to all of , as well as connections to subsequent commitment-security results in the literature.

Abstract

Committing to information is a central task in cryptography, where a party (typically called a prover) stores a piece of information (e.g., a bit string) with the promise of not changing it. This information can be accessed by another party (typically called the verifier), who can later learn the information and verify that it was not meddled with. Merkle trees are a well-known construction for doing so in a succinct manner, in which the verifier can learn any part of the information by receiving a short proof from the honest prover. Despite its significance in classical cryptography, there was no quantum analog of the Merkle tree. A direct generalization using the Quantum Random Oracle Model (QROM) does not seem to be secure. In this work, we propose the quantum Merkle tree. It is based on what we call the Quantum Haar Random Oracle Model (QHROM). In QHROM, both the prover and the verifier have access to a Haar random quantum oracle and its inverse. Using the quantum Merkle tree, we propose a succinct quantum argument for the Gap--Local-Hamiltonian problem. Assuming the Quantum PCP conjecture is true, this succinct argument extends to all of QMA. This work raises a number of interesting open research problems.
Paper Structure (16 sections, 3 theorems, 5 equations, 4 figures, 2 algorithms)

This paper contains 16 sections, 3 theorems, 5 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Merkle Tree Algorithm
  3. A Failed Attempt to Adapt Merkle Tree in the Quantum Setting
  4. The Quantum Haar Random Oracle Model ($\mathsf{QHROM}$) and Quantum Merkle Tree
  5. A Candidate for Succinct Quantum Argument for Gap-$k$-$\mathsf{LH}$ in $\mathsf{QHROM}$
  6. Open Questions and Follow-up Works
  7. Preliminaries
  8. Notation
  9. The Quantum Haar Random Oracle Model
  10. Quantum Local Proofs
  11. The Quantum $\mathsf{PCP}\xspace$ Conjecture
  12. A Candidate Succinct Argument for $\mathsf{LocalQMA}$ in $\mathsf{QHROM}$
  13. The Succinct Protocol $\Pi_{\sf succinct}$
  14. Notation.
  15. Analysis of $\Pi_{\sf succinct}$
  16. ...and 1 more sections

Key Result

Corollary 1

If $\mathsf{QPCP}$ holds, then there are constants $k,\gamma \in \mathbb{N}$ and $c,s \in [0,1]$ satisfying that $s < c$, such that

Figures (4)

  • Figure 1: An illustration of the toy example for the classical Merkle tree
  • Figure 2: An illustration of the toy example for the quantum Merkle tree
  • Figure 3: An illustration of the quantum Merkle tree with $\ell = 2^{d}$ input blocks; when block $x^{(2)}$ is requested by $\mathcal{V}$, $\mathcal{P}$ sends all the diamond shape nodes.
  • Figure 4: An illustration of the labeling of the nodes in the tree $T_\ell$ with $\ell$ leaves

Theorems & Definitions (10)

  • Definition 1
  • Conjecture 1
  • Definition 2
  • Definition 3: $(k,\gamma)$-$\mathsf{LocalQMA}$
  • Conjecture 2: $\mathsf{QPCP}$ conjecture
  • Corollary 1
  • Theorem 3.1
  • proof
  • Conjecture 3: $\Pi_{\sf succinct}$ is sound in $\mathsf{QHROM}$
  • Corollary 2