Quantum Auction: Quantum-Enhanced Bidding & Security

• Quantum auction is a family of protocols that leverage quantum states and entanglement to improve bid communication, privacy, and verification in auction settings.
• One approach utilizes quantum-enhanced communication to overcome bidding constraints, as shown by improved success probabilities in Bridge bidding using entangled photon pairs.
• Other strands integrate quantum cryptographic techniques and blockchain architectures to secure sealed-bid auctions and compress communication in classical mechanism designs.

Searching arXiv for relevant papers on quantum auctions and related quantum market mechanisms. Quantum auction is not a single protocol class but a family of constructions in which quantum resources are used to improve some part of auction-like interaction: constrained bidding communication, sealed-bid privacy, verifiable winner determination, communication-efficient implementation of classical revenue-maximizing mechanisms, or secure market-clearing infrastructure. In the literature considered here, the term covers at least four distinct regimes: quantum-enhanced communication inside the auction phase of Bridge, quantum cryptographic sealed-bid auction protocols, quantum communication models for classical mechanism design, and quantum-secured or quantum-accelerated market computation for auction-like markets (Muhammad et al., 2014, Sharma et al., 2016, Rubinstein et al., 2023, Huang et al., 2024).

1. Terminology and scope

The broadest unifying feature is that private economic or game-theoretic information is processed under communication, privacy, or verification constraints, and quantum states, entanglement, or quantum-secured channels are used to alter those constraints. This does not imply a uniform mechanism-design agenda. Some works treat “auction” literally as a sealed-bid mechanism with private bids and winner selection; others study bidding phases in games such as Bridge; still others analyze communication complexity of implementing already classical Bayesian auctions, or use quantum infrastructure to secure otherwise classical market-clearing layers (Muhammad et al., 2014, Sun et al., 2020, Rubinstein et al., 2023, Zhu, 16 Oct 2025).

A central distinction is between quantum auction as cryptographic protocol and quantum auction as economic computation. In cryptographic work, the emphasis is bid privacy, binding, fairness, collusion resistance, and verifiability. In communication-complexity work, the emphasis is how many qubits or bits are required to implement an incentive-compatible auction. In market-algorithm work, the emphasis is equilibrium computation, price discovery, or settlement under quantum-secured infrastructure rather than sealed-bid privacy per se (Sharma et al., 2016, Rubinstein et al., 2023, Huang et al., 2024).

This multiplicity of meanings also explains a recurring clarification in the literature: several papers explicitly state that their contribution is not a new auction rule in the classical economic sense, but a new communication or security architecture embedded in an auction-like setting. That clarification is explicit for Bridge bidding and for quantum-secured energy trading, and it is implicit in work that uses quantum communication only to compress or secure implementation of a pre-existing auction objective (Muhammad et al., 2014, Zhu, 16 Oct 2025).

2. Quantum-enhanced bidding as constrained communication

A canonical example of the communication-complexity view is "Quantum bidding in Bridge" (Muhammad et al., 2014). The paper does not redesign an economic auction; it studies the auction phase of Bridge, where partners communicate under strict bidding constraints. In the Roman Key-card Blackwood subproblem analyzed there, one partner holds two relevant bits of information—key-card information and trump-queen information—but can safely communicate only one bit through the allowed bid. That subproblem is formalized as a one-way task in which Alice has $(a_0,a_1)$, Bob has $b \in \{0,1\}$, Alice sends one bit $m$, Bob outputs $R$, and success means $R=a_b$.

The paper identifies the Bridge task with a biased nonlocal Clauser-Horne-Shimony-Holt game and equivalently with a $2 \to 1$ entanglement-assisted random access code. The classical optimum is obtained by committing in advance to sending whichever bit is more valuable on average, while the unsent bit is guessed from its marginal distribution. The quantum protocol instead uses shared entanglement, Alice’s setting $a=a_0\oplus a_1$, Bob’s setting $b$, Alice’s message $m=A\oplus a_0$, and Bob’s output $R=B\oplus m$, so that $b \in \{0,1\}$0 whenever $b \in \{0,1\}$1. The maximal quantum success probability is

$b \in \{0,1\}$2

The gain is experimentally modest but explicit in the Bridge-relevant regime. Using expert estimates with $b \in \{0,1\}$3, $b \in \{0,1\}$4, hence $b \in \{0,1\}$5, and $b \in \{0,1\}$6, the paper reports classical success probability $b \in \{0,1\}$7 and quantum success probability $b \in \{0,1\}$8. In the symmetric case $b \in \{0,1\}$9, it reports $m$0 against a classical $m$1 for $m$2, and for $m$3 it reports theoretical quantum $m$4, experimental quantum $m$5, and classical $m$6. The implementation uses polarization-entangled photons in the Bell state

$m$7

realized physically as $m$8, with type-II spontaneous parametric down-conversion in a $m$9 BBO crystal, variable-ratio beam splitters to realize biased inputs, and single-photon detection over $R$0 per setting.

Two further features make this case distinctive. First, the overt communication does not increase: one bid is still sent, and the advantage comes from shared entanglement and local measurements rather than from forbidden signaling. Second, the authors note that Bridge rules permit auxiliary communication about detection events, allowing the players to repeat trials until both detect particles; this softens the detector-efficiency bottleneck relative to standard communication-complexity demonstrations. The paper is also explicit that the result applies only to certain clearly defined phases of Bridge, not to the game as a whole.

3. Sealed-bid auction protocols and cryptographic architectures

A large portion of the quantum-auction literature concerns sealed-bid cryptography. Here the object is not better price formation but private bid submission, winner determination, and resistance to cheating coalitions. One design line uses multipartite entanglement and structured message passing. "Quantum sealed-bid auction using a modified scheme for multiparty circular quantum key agreement" (Sharma et al., 2016) arranges the auctioneer and bidders in a circle and then partitions the circle into $R$1 sub-circles. If there are $R$2 bidders, the paper defines

$R$3

In each initiator’s round, the initiator belongs to all $R$4 sub-circles and every other participant belongs to exactly one. Each participant prepares $R$5 copies of a $R$6-partite entangled state, inserts decoy qubits into the travel sequences, and the parties encode bid bits via disjoint order-2 subgroups $R$7, with $R$8 encoded by $R$9 and $R=a_b$0 by the participant’s assigned nontrivial unitary. The point of the sub-circle topology is structural: it limits collusion visibility, and the paper states that if fewer than $R=a_b$1 attackers collude, they cannot cheat the remaining bidders and the auctioneer. Increasing $R=a_b$2 simultaneously increases collusion resistance and reduces the size of each entangled state, which the paper presents as a rare case where security and implementability improve together.

A different design line replaces entanglement in the main bid channel by single photons. "Multiparty Quantum Sealed-Bid Auction Using Single Photons as Message Carrier" (Liu et al., 2020) has the auctioneer prepare random single-photon sequences from $R=a_b$3, with bid bits encoded by

$R=a_b$4

Decoy photons are used for channel checking, while fairness is delegated to a post-confirmation phase based on EPR pairs and a permutation operator. Each bidder encodes two-bit blocks of the bid into Bell states and permutes the particles before sending them to other bidders; after the auctioneer announces the winner, the winner reveals the permutation so others can reconstruct the EPR ordering and Bell-measure the announced winning bid. The paper’s stated novelty is that the main auction path uses only single photons and single-particle measurements, while entanglement is retained only for fairness auditing.

Subsequent work subjected such proposals to cryptanalysis. "Quantum and semi-quantum sealed-bid auction: Vulnerabilities and advantages" (Asagodu et al., 2021) analyzes the protocols of Liu et al. and Zhang et al. and argues that several claims are incorrect and that the protocols are vulnerable to participant attacks, outsider attacks, and collusion. For Zhang et al.’s first protocol, which encoded post-confirmation data in $R=a_b$5 and $R=a_b$6, the paper derives a single-copy bit-recovery success probability

$R=a_b$7

and then shows that intercepting multiple copies makes recovery highly reliable. It proposes a genuinely semi-quantum replacement in which only the auctioneer has full quantum capability. Bidders are restricted to semi-quantum actions such as reflection and preparation of computational-basis qubits, and the post-confirmation step is classical: bidder $R=a_b$8 sends each other bidder $R=a_b$9

$2 \to 1$0

so that after the auctioneer announces winner $2 \to 1$1, every bidder checks consistency through $2 \to 1$2. This shifts the design from quantum post-confirmation to key-and-hash commitment with semi-quantum key distribution.

A separate strand integrates sealed-bid auctions with distributed ledgers. "Bit Commitment for Lottery and Auction on Quantum Blockchain" (Sun et al., 2020) uses quantum bit commitment and a quantum blockchain to realize a decentralized sealed-bid auction with five phases: bidding, opening, decision, verification, and publication. Each buyer commits the bid to the seller and all miners; each buyer opens the bid to the seller; the seller selects the highest bid; each miner verifies the seller’s claim using the winning bid and a permuted list of losing bids $2 \to 1$3, checking

$2 \to 1$4

and the miners then reach consensus and add the result to the blockchain. The paper’s stated security properties are bid privacy, posterior privacy, bids’ binding, decentralization, and unconditional security, but it also leaves major components underspecified, including settlement, bidder authentication, collusion analysis, and exact integer commitment mechanics.

More recent work pushes toward one-shot submission. "New Quantum Internet Applications via Verifiable One-Time Programs" (Stambler, 26 Sep 2025) introduces Ver-OTPs and Open Secure Computation, then instantiates a single-round sealed-bid auction. Each bidder sends a single message containing an MHE ciphertext of the auction input and a Ver-OTP that can be checked for consistency before use and then consumed exactly once to provide partial decryption. The auction function can be specialized so that bidder $2 \to 1$5’s input is

$2 \to 1$6

and the auction output is the highest bid $2 \to 1$7 together with

$2 \to 1$8

Because the OSC functionality allows the evaluator to partition inputs, the auction sketch adds a majority condition over registered bidders so that the function outputs $2 \to 1$9 unless a majority of registered inputs are present. The paper is explicit that this is an application sketch built on the formal OSC theorem, not a full end-to-end auction deployment proof.

A structural reframing appears in "On the generic structures of the protocols for quantum auction and quantum summation and their relation" (Sandhu et al., 26 Jun 2026). The paper argues that many auction primitives reduce to repeated calls to a summation oracle. For threshold $a=a_0\oplus a_1$0, define

$a=a_0\oplus a_1$1

Then $a=a_0\oplus a_1$2 iff some bid exceeds threshold $a=a_0\oplus a_1$3, so repeated evaluations recover the maximum bid, and a winner can then be found via a winner-indicator function and Grover search. The converse reduction interprets summation itself as an auction-like amplitude-estimation task. The paper also makes the leakage structure explicit: if the full threshold transcript is revealed, the exact bid histogram is reconstructed through $a=a_0\oplus a_1$4, so privacy depends on what transcript is exposed rather than on quantumness alone.

4. Quantum communication complexity of classical auctions

A distinct line of work does not alter the auction objective at all; it asks how much communication is needed to implement a classical Bayesian auction when quantum messages are allowed. "Quantum Communication Complexity of Classical Auctions" (Rubinstein et al., 2023) studies single-buyer, multi-item Bayesian revenue maximization with a strategic buyer and a seller who may exchange qubits and classical bits under incentive constraints.

For approximately optimal auctions, the main positive theorem says that if a mechanism can allocate only one of $a=a_0\oplus a_1$5 possible bundles, then there is an incentive-compatible quantum auction protocol that guarantees a $a=a_0\oplus a_1$6-fraction of that mechanism’s expected revenue using $a=a_0\oplus a_1$7 qubits in expectation. This yields $a=a_0\oplus a_1$8 expected qubits for unit-demand buyers and $a=a_0\oplus a_1$9 for arbitrary combinatorial valuations. The compression comes from encoding a lottery over $b$0 outcomes as an amplitude state

$b$1

but the protocol must add a low-probability verification branch and very large off-path penalties to keep a strategic buyer from deviating.

That caveat is not cosmetic. If $b$2 is expected quantum communication and $b$3 bounds payments, the paper proves lower bounds on the product $b$4: for unit-demand valuations, $b$5; for gross substitutes, $b$6; and for XOS valuations, $b$7. The paper’s interpretation is that exponentially compressed communication can require exponentially large punishments of deviating buyers.

For exact optimal auctions, the paper gives three fine-grained separations even in the two-item additive setting. First, there exists a prior for which no finite classical auction protocol achieves optimal revenue, but a one-way quantum protocol using one qubit and two classical bits does. Second, there exists a prior for which no finite one-way quantum protocol is optimal, but a barely interactive protocol becomes optimal when the seller first sends one qubit of an EPR pair and the buyer returns one qubit and two classical bits. Third, there exists a prior for which no finite-communication multi-round quantum protocol can implement the exact optimum at all. The negative result is tied to semialgebraicity: finite quantum protocols induce buyer-utility functions that are semialgebraic, while some optimal utilities involve non-semialgebraic forms such as Lambert $b$8. The resulting picture is not monotone “quantum always helps,” but a hierarchy separating classical, one-way quantum, interactive quantum, and unrestricted finite quantum communication models.

5. Market equilibrium computation and quantum-secured trading systems

Another meaning of quantum auction concerns auction-like markets rather than sealed-bid winner determination. "Quantum algorithm for large-scale market equilibrium computation" (Huang et al., 2024) studies linear Fisher markets with $b$9 buyers, $m=A\oplus a_0$0 divisible goods, budgets $m=A\oplus a_0$1, and linear utilities

$m=A\oplus a_0$2

The paper formulates equilibrium through the Eisenberg–Gale program and a bid-based Shmyrev program, then develops a quantum version of proportional response via a faulty proportional response variant that tolerates multiplicative estimation error. The key claim is the first quantum algorithm for market equilibrium computation with sublinear performance in the product $m=A\oplus a_0$3: classical proportional response has runtime $m=A\oplus a_0$4, while the quantum algorithm achieves

$m=A\oplus a_0$5

up to logarithmic factors, with QRAM space $m=A\oplus a_0$6. The output is query access to the approximate bid matrix rather than explicit materialization of the full $m=A\oplus a_0$7 object. The paper is explicit that this is not a sealed-bid auction protocol, but it is directly relevant to auction markets, budget pacing, recommender systems, and fair division.

"Q-EnergyDEX: A Zero-Trust Distributed Energy Trading Framework Driven by Quantum Key Distribution and Blockchain" (Zhu, 16 Oct 2025) uses the term quantum in still another sense. The auction and market-clearing layer is classical: a Stackelberg-constrained bilateral auction in which a grid operator chooses network prices $m=A\oplus a_0$8 and each prosumer solves

$m=A\oplus a_0$9

What is quantum is the surrounding security stack: a QKD-driven key-management service, the Q-SAH authentication protocol, and the PoR-Lite blockchain consensus mechanism. The availability of secure auction execution is explicitly coupled to entropy supply through formulas such as

$R=B\oplus m$0

for key generation rate and

$R=B\oplus m$1

for key-pool dynamics. The paper therefore treats auction feasibility as conditioned by cryptographic resource availability rather than by a quantum algorithm for market clearing.

"Theory of Quantum Games and Quantum Economic Behavior" (Ikeda et al., 2020) is not an auction paper in the narrow sense, but it is foundational for quantum mechanism design. It models a quantum economy with quantum goods and entangled strategies, proves a convergence theorem under which repeated exchange drives quantum commodity states toward a classical mixed state, and exhibits examples where a player can obtain a desired transfer regardless of another player’s inclination because of noncommutation with an entangling operator. The paper states that such properties “shed new light on theories of mechanism design, auction and contract in the quantum era.” This suggests a mechanism-design interpretation in which allocation feasibility, collusion, and consent constraints may change when strategy spaces are operator-valued and entangled, although the paper itself does not formulate an auction theorem.

6. Security models, leakage, and recurring limitations

Across the literature, the hardest problems are not raw winner selection but leakage control, collusion, and robustness under strategic deviation. Several protocols explicitly assume authenticated classical channels, practical decoy-state checks, or restricted corruption models. The circular MQKA-based architecture relies on structural anti-collusion and decoy qubits rather than composable proofs (Sharma et al., 2016). The semi-quantum auction work is attack-based and depends on one-way hash functions in its post-confirmation phase (Asagodu et al., 2021). The Ver-OTP/OSC framework requires a CRS, a strengthened multi-key homomorphic encryption primitive, and a receiver hardware assumption, and it still assumes the auctioneer will post the result (Stambler, 26 Sep 2025).

Privacy is also subtler than “sealed bid” language may suggest. In the quantum-blockchain auction, posterior privacy means that losing bids are published only as a permuted list, so losing values remain visible even if identities are hidden (Sun et al., 2020). In the summation-based reduction, exact threshold transcripts can reveal the whole bid histogram through $R=B\oplus m$2, leaving only bidder-to-value assignment hidden (Sandhu et al., 26 Jun 2026). In Bridge bidding, the public bid itself becomes less exploitable because it is informative only when combined with Bob’s local measurement outcome, but the paper does not claim full secrecy against all inference channels (Muhammad et al., 2014).

A second recurring limitation is scope. The Bridge result is about a narrowly defined slam-investigation bottleneck, not “quantum Bridge” in general (Muhammad et al., 2014). The communication-complexity paper treats a single-buyer Bayesian seller problem, not multi-bidder auction markets (Rubinstein et al., 2023). The market-equilibrium algorithm applies to Fisher markets with divisible goods and QRAM access, not to general combinatorial auctions (Huang et al., 2024). The quantum-blockchain and energy-trading papers secure or decentralize the auction environment, but do not provide fully specified settlement, identity, or collusion-resistant auction semantics (Sun et al., 2020, Zhu, 16 Oct 2025).

A common misconception is therefore that quantum auction denotes a single mature mechanism class. The literature instead supports a more granular view. One strand shows that entanglement can make a fixed one-bit bid more useful without increasing overt communication. Another constructs sealed-bid protocols with varying mixes of entanglement, single photons, semi-quantum users, blockchain, and one-time programs. A third shows that quantum communication can compress the implementation of classical Bayesian auctions but only under delicate incentive constraints. A fourth uses quantum security or quantum algorithms to support market-clearing systems that remain economically classical. Taken together, these works indicate that “quantum auction” is best treated as an umbrella term for quantum-enhanced communication, privacy, verification, and computation around bidding and market-clearing tasks, rather than as a single canonical auction model.