Unentangled Quantum Interactive Proofs

• Unentangled quantum interactive proofs are models where provers send messages that remain separate from any private workspace, ensuring no hidden entanglement.
• They use entanglement-breaking channels and post-measurement branching to simulate powerful verification schemes, achieving exponential computational gains as seen in QIP[3] equating to NEXP.
• This framework contrasts classical and fully entangled protocols, opening new avenues in protocol design and complexity theory while mitigating cheating via entanglement.

Unentangled quantum interactive proofs are models of quantum verification protocols in which the messages sent by the prover (or provers) are always guaranteed to be unentangled from their private workspace, or—more generally—where multiple provers are constrained to give unentangled proofs. This restriction profoundly influences the computational power and structure of quantum interactive proof classes. The defining features of these models are the "entanglement-breaking" property of prover channels and the resulting consequences for expressivity, soundness, and protocol design, especially in contrast to both classical and fully entangled quantum protocols.

1. Unentanglement as a Computational Resource

The central premise of unentangled quantum interactive proofs is that the prover, or provers, are restricted to sending messages that are not entangled with any private workspace or with one another. Formally, this restriction is enforced by requiring the prover’s channel to be "entanglement-breaking", that is, every action is of the form: $$\Phi(\rho) = \sum_\ell \operatorname{Tr}(E_\ell \rho) | \phi_\ell \rangle \langle \phi_\ell |$$ where $\{E_\ell\}$ form a POVM and $|\phi_\ell\rangle$ are fixed states. For multi-prover systems, the initial proofs are required to be fully separable (unentangled) across the provers; for example, in QMA(2) or QMA(k), the witness states are tensor products $\rho_1 \otimes \cdots \otimes \rho_k$.

This structural constraint limits the strategies available to potentially dishonest provers—entanglement, a powerful resource for cheating in interactive and nonlocal games, is now disallowed. Nevertheless, and perhaps counterintuitively, unentanglement dramatically alters the expressive power of certain quantum proof systems.

2. Expressive Power Jump: QIP[3] and NEXP

A major finding is that in the three-round setting, imposing the unentanglement constraint on QIP3 results in exponential increases of computational power: $\text{Unentangled-QIP[3]} = \text{NEXP}$ whereas standard (possibly entangled) QIP[3] equals PSPACE (Grewal et al., 18 Sep 2025).

The underlying intuition is that, in unentangled QIP[3], the protocol avoids the prover "hiding" information in entanglement and enables the simulation of quantum PCP-type protocols for NEXP. Specifically, the first message (quantum witness) is not entangled with the prover’s private register. After a measurement by the verifier (which can be adaptively branched, i.e., with post-measurement branching), the protocol proceeds to extract parts of an exponentially long classical proof, thus inheriting the power of NEXP.

Crucially, the jump in expressive power is due to the lack of entanglement between the prover’s sent message and his local workspace, not simply between sent messages.

3. Post-Measurement Branching: Adaptive Verification

"Post-measurement branching" refers to the verifier's ability to adapt their subsequent protocol steps based on the outcome of an intermediate measurement, conditioning both on the classical result and the residual quantum state. This quantum adaptivity is not mirrored in classical interactive proofs, where public-coin (Arthur–Merlin) and private-coin (IP) protocols coincide in expressive power for constant rounds.

In the quantum setting:

• If the verifier uses post-measurement branching (i.e., can adapt the challenge to the measurement outcome), the unentangled protocol can achieve the full power of NEXP in three rounds.
• If post-measurement branching is prohibited (i.e., the verifier's challenge depends only on public randomness, not the result of a quantum measurement), the power collapses to that of QAM (Quantum Arthur–Merlin), which is believed to be much weaker: $\text{Unentangled-QMACM} \subseteq \text{QAM}$ (Grewal et al., 18 Sep 2025).

This shows a sharp distinction from classical interactive proofs (where IP[2] = AM), highlighting that quantum post-measurement adaptivity is a uniquely powerful resource for verification.

4. Complexities, Separations, and Examples

The separation between protocols with and without post-measurement branching is further exemplified in the two-round quantum-classical setting:

• In QCAM (quantum-classical Arthur–Merlin, where challenge is generated by public randomness), the class matches BP·QCMA.
• With general post-measurement adaptivity (QCIP[2]), the verification power can be as large as BQP^NP^PP, strictly stronger than QCAM.

A key technical observation is that, unlike in classical protocols, the ability to "branch" adaptively on measurement outcomes and use residual quantum states cannot be efficiently simulated by a non-adaptive, public-coin approach.

This yields provable hierarchies: $$\text{QCAM} = \text{BP} \cdot \text{QCMA} \subseteq \text{BQ} \cdot \text{QCMA} \subseteq \text{QCIP[2]} \subseteq \text{BQP}^{\text{NP}^{\text{PP}}}$$ (Grewal et al., 18 Sep 2025).

5. Interactions with Other Quantum Proof Models

The paper of unentangled quantum interactive proofs interfaces intricately with other quantum proof models. In QMA(2) (quantum Merlin–Arthur with two unentangled proofs), "product" and "swap" tests have been central tools for verifying separability (Gutoski et al., 2013, Jeronimo et al., 23 Feb 2024). The construction of dimension-independent disentangler channels (Jeronimo et al., 23 Feb 2024) allows amplifying k-partite separability from bipartite unentangled inputs, facilitating robust gap amplification and capturing large complexity classes (including NEXP) even with proofs having nearly general real amplitudes.

Notably, the ability to enforce or test unentanglement makes quantum interactive proofs robust against a class of adversarial strategies that would otherwise exploit entanglement for cheating or coordination.

6. Implications, Open Questions, and Future Directions

Imposing unentanglement (and leveraging post-measurement branching) gives rise to a family of quantum proof systems with complexity-theoretic power exceeding that of their unconstrained, entangled analogues. This runs counter to early intuition, as entanglement is often viewed as a resource for increasing computational or verification power.

Key directions for further investigation include:

• Determining the precise boundaries and potential collapse of quantum interactive proof hierarchies with more rounds, both in the presence and absence of post-measurement branching.
• Characterizing intermediate classes between QMA(2) and unentangled QIP[3], and identifying practical verification problems that populate these boundaries.
• Understanding relativized separations between QAM, QIP[3], and BQP-oracle variants in the presence of unentanglement and branching resources.
• Exploring the interaction between unentanglement constraints and classical cryptographic reductions, especially in device-independent quantum cryptography and delegated verification.

The emerging message is that, in the quantum landscape, removing entanglement from the prover’s channel—especially in conjunction with post-measurement branching—can paradoxically enhance the expressiveness of quantum interactive proofs, and that the subtlety of quantum adaptivity distinguishes these models sharply from their classical counterparts (Grewal et al., 18 Sep 2025).