Proof of Quality (PoQ)

• Proof of Quality (PoQ) is a framework that combines decentralized consensus, cryptographic proofs, and meta-complexity to verify computational quality and demonstrate quantum advantage.
• It employs cost-aware reward mechanisms in decentralized AI inference, using scaled evaluator scores to incentivize both output quality and efficiency.
• PoQ protocols span practical LLM inference systems to quantum proofs of quantumness, highlighting trade-offs, limitations, and open research challenges.

Proof of Quality (PoQ) encompasses several technically distinct paradigms across cryptography, computational verification, and decentralized AI inference that establish trust in outputs or computational advantage through consensus, cryptographic protocols, or complexity-theoretic gaps. The notion originated in the context of consensus-driven quality assessment for generative AI model inference on blockchains, but the term is also foundational in the cryptographic literature as “proof of quantumness,” denoting protocols by which a prover demonstrates the capacity for quantum or otherwise hard-to-verify computation to a classical or semi-efficient verifier. PoQ mechanisms replace full cryptographic verification of process or execution trace with verifiable guarantees about output quality or infeasibility of classical replication, adopting evaluation, incentive, or challenge–response paradigms tailored to their underlying domain.

1. Decentralized LLM Inference and PoQ Consensus

In decentralized AI inference, PoQ provides a scalable alternative to cryptographic proof formats such as SNARKs or optimistic re-execution by delegating verification to consensus on the quality of outputs. Formally, in a PoQ system for LLM inference, an inference node (F-node) submits the result $r=F(q)$ for input $q$; multiple lightweight evaluator nodes (M-nodes) individually score $r$; these scores are aggregated to form a consensus quality score $Q$, which drives the system’s reward allocation. Critically, reward mechanisms are aligned with both output quality and computational efficiency: PoQ employs cost-aware formulas to normalize both quality (e.g., min–max scaling evaluator scores $e_{m,f,i}\in[0,10]$) and cost (latency-based $C_F(f)$ or $C_M(m)$), combining these via linear functions: $R_F(i,f) = \alpha_F\,Q_{i,f} - \beta_F\,C_F(f)$

$$R_M(i,f,m) = \alpha_M\,C^{\mathrm{close}}_{i,f,m} - \beta_M\,C_M(m)$$

with tunable coefficients. This ensures that high-quality, low-cost inference and evaluation are incentivized, penalizing latency and low alignment.

The PoQ evaluation pipeline incorporates multiple quality signals: ground-truth metrics (e.g., token-F1), lightweight learned evaluators (cross-encoder and bi-encoder models), and GPT-based meta-judgments. Only the normalized signals from learned evaluators directly impact PoQ on-chain rewards, but ground truth and GPT-4 judgments guide choice and calibration of evaluator architectures. Notably, semantic textual similarity bi-encoders (e.g., STS-DistilRoBERTa) exhibit higher correlation ($\approx0.66$ with F1, $0.29$ with GPT) than cross-encoders, justifying their prioritization. Simulation studies over thousands of PoQ rounds confirm that this cost-aware scheme rewards both inference and evaluator nodes in direct proportion to efficiency and accuracy, supporting economic sustainability in decentralized AI services (Tian et al., 18 Dec 2025).

2. PoQ in Cryptographic Verification: Proofs of Quantumness

In quantum cryptography, PoQ refers to “proofs of quantumness”: interactive protocols whereby a (potentially resource-limited) verifier tests that a prover executed a computation infeasible for any classical device. This setting models a two-party protocol $(V,P)$ with completeness (honest quantum $P_Q$ can convince $V$) and soundness guarantees (no classical $P_C$ with bounded resources convinces $V$ above a negligible threshold), formalized as: $$\text{Pr}[V\langle P_Q\rangle \text{ accepts}] \geq c(n)$$

$$\text{Pr}[V\langle P_C\rangle \text{ accepts}] \leq s(n)$$

with a non-negligible gap $c(n)-s(n)$. Provably secure PoQ protocols under bounded-storage models have been realized, relying on hardness results for classical memory-bounded adversaries (e.g., Raz’s learning parity with noise lower bound). Two canonical protocols have been constructed:

• Quadratic Gap: Based on linear mappings and mod 2 arithmetic, achieving a $(n+2)^2$ vs. $n^2/20$ quantum/classical storage gap, feasible for near-term qubit devices.
• Exponential Gap: Employs streamed randomness, interactive hashing, and coherent state “stitching” for exponential advantage in quantum resource efficiency versus classical adversary storage, at the expense of deeper prover circuits.

Both protocols require only minimal quantum gates (Hadamard, CNOT, phase gates), efficient classical verifier computation, and no cryptographic assumptions beyond bounded storage (Malavolta et al., 29 May 2025).

3. Meta-Complexity and the Foundations of Proofs of Quantumness

PoQ, especially in its inefficient-verifier variant (IV-PoQ), is now established to be equivalent in power to classically-secure one-way puzzles (OWPuzzs), a primitive strictly weaker than one-way functions. Recent work demonstrates that IV-PoQ protocols can be constructed from meta-complexity-theoretic assumptions, for instance, the average-case hardness of the $\mathsf{GapK}[s_1,s_2]$ problem—that distinguishing between strings of low and high Kolmogorov complexity is intractable on average for classical adversaries. If GapK is weakly-classical-average-hard, one constructs a classically-secure one-way puzzle, which directly yields an IV-PoQ: the verifier samples a puzzle instance, the quantum prover inverts it (via quantum search or enumeration), and an unbounded-time verifier checks correctness.

The formal equivalence (IV-PoQ $\iff$ OWPuzz) reveals that quantum advantage protocols—a quantum prover outperforming any classical PPT adversary in interactive or non-interactive settings—can be based on the weakest known cryptographic or complexity-theoretic assumptions, below OWFs, encompassing e.g., PRSGs, OWSGs, and pseudorandom unitaries. This foundational result implies that absence of quantum advantage would collapse a wide array of quantum-only cryptographic primitives (Hiroka et al., 2 Oct 2024, Morimae et al., 1 Oct 2024).

4. Protocol Architectures and Instantiations

Representative PoQ and IV-PoQ protocols adhere to several recurring structures:

• Quality-Consensus LLM PoQ: A user query is handled by a designated inference node; $K$ validator nodes perform lightweight quality assessment; results are aggregated on-chain. Rewards are assigned by consensus formulas (e.g., exponential decay, deviation penalties, or linear quality–cost trades), and adversarial incentives are mitigated by game-theoretic design. Evaluator model selection is empirical (correlation with ground truth or human proxies). System architectures such as PQML implement these ideas for large-scale generative AI (Zhang et al., 28 May 2024).
• Bounded-Storage Quantum PoQ: Protocols generate superpositions and “claw” states with classical-quantum interaction sequences, using challenge–response phases mimicking cryptographic commitment and randomness extraction. Completeness and soundness are derived from memory-bounded learning hardness. Protocol depth and verifier complexity are tunable trade-offs (Malavolta et al., 29 May 2025).
• Commitment–based IV-PoQ: Built from statistically hiding, computationally binding commitments reducible to OWF assumption, these protocols combine commitment transcripts, pairwise-independent hashing, and classical postprocessing. Completeness is ensured by a quantum prover’s ability to collapse to a two-term superposition; soundness follows from adversarial inability to extract both preimage states, protected by computational binding (Morimae et al., 2023, Morimae et al., 1 Oct 2024).
• Non-interactive IV-PoQ (Quantum Samplers): For sampling- or search-based quantum advantage, the verifier tasks the quantum device with outputting hard-to-classically-sample distributions (e.g., IQP circuit outcomes), later checked (inefficiently) by Kolmogorov complexity estimation or direct solution verification (e.g., factoring).

5. Economic and Security Analysis in PoQ Systems

In decentralized inference, PoQ system security does not rest on cryptographic soundness, but rather on rationality and economic incentives. The cost-aware PoQ framework employs tunable coefficients to regulate trade-offs between quality and computational cost. Simulation confirms these schemes can sustain marketplaces where both inference and evaluation services are efficiently priced—nodes with higher “quality-per-cost” are disproportionately compensated, and low-quality or inefficient participants are penalized. The system achieves rapid consensus (order 10–100 ms), in contrast to orders-of-magnitude slower cryptography-dependent approaches (minutes–hours for ZKML or OPML).

Threat models in blockchain PoQ assume rational (profit-maximizing) adversaries, not arbitrary Byzantine threats. Empirical diversity in evaluator outputs (variance in quality scorers) is required to deter validator collusion or gaming. Performance analysis exhibits that choice of validator model architecture directly impacts both security and throughput, with bi-encoders proving most effective (Tian et al., 18 Dec 2025, Zhang et al., 28 May 2024).

6. Connections, Limitations, and Open Problems

PoQ research has revealed several critical connections and limitations:

• Expressiveness: PoQ covers a broad landscape, from practical decentralized AI consensus (where the focus is scalable, rapid verification by replicated evaluation) to formal cryptographic quantification of computational advantage.
• Limitations: PoQ protocols for LLM inference do not cryptographically guarantee traceability of the exact inference steps—only that the consensus over outputs is honest in expectation under rationality. Cryptographic IV-PoQ protocols often rely on hardness assumptions not yet reducible to worst-case complexity, and efficiently verifiable PoQ without any quantum-verifier (QPT) step is tightly characterized by the strength of available one-way puzzles.
• Open Problems: Open directions include separating or unifying zero-knowledge IV-PoQ variants, further weakening the base assumptions for quantum advantage, and exploring trade-offs between soundness, rounds, and efficiency across both practical and complexity-theoretic domains.

In summary, PoQ unifies practical output-verification protocols for decentralized AI, rigorous challenge–response cryptographic schemes for quantum verification, and meta-complexity foundations for quantum advantage, providing a multidimensional framework with both practical and theoretical impact (Tian et al., 18 Dec 2025, Malavolta et al., 29 May 2025, Morimae et al., 1 Oct 2024, Hiroka et al., 2 Oct 2024, Morimae et al., 2023, Zhang et al., 28 May 2024).