###### Abstract  
In this work, we present a new protocol that accomplishes multiparty quantum private comparison leveraging maximally entangled \(\left|GHZ_{3}\right\rangle\) triplets. Our intention was to develop a protocol that can be readily executed by contemporary quantum computers. This is possible because the protocol uses only \(\left|GHZ_{3}\right\rangle\) triplets, irrespective of the number \(n\) of millionaires. Although it is feasible to prepare multiparticle entangled states of high complexity, this is overly demanding on a contemporary quantum apparatus, especially in situations involving multiple entities. By relying exclusively on \(\left|GHZ_{3}\right\rangle\) states, we avoid these drawbacks and take a decisive step toward the practical implementation of the protocol. An important quantitative characteristic of the protocol is that the required quantum resources are linear both in the number of millionaires and the amount of information to be compared. Additionally, our protocol is suitable for both parallel and sequential execution. Ideally, its execution is envisioned to take place in parallel. Nonetheless, it is also possible to be implemented sequentially if the quantum resources are insufficient. Notably, our protocol involves two third parties, as opposed to a single third party in the majority of similar protocols. Trent, commonly featured in previous multiparty protocols, is now accompanied by Sophia. This dual setup allows for the simultaneous processing of all \(n\) millionaires' fortunes. The new protocol does not rely on a quantum signature scheme or pre-shared keys, reducing complexity and cost. Implementation wise, uniformity is ensured as all millionaires use similar private circuits composed of Hadamard and CNOT gates. Lastly, the protocol is information-theoretically secure, preventing outside parties from learning about fortunes or inside players from knowing each other's secret numbers. quantum private equality comparison; multiparty quantum private comparison; \(\left|GHZ_{3}\right\rangle\) states; semi-honest third party; quantum entanglement; quantum cryptography; quantum games +
Footnote †: journal:  
One approach, often referred to as post-quantum or quantum-resistant cryptography, advocates for reliance on problems that are believed to be, but not yet formally proven to be, computationally hard. The other approach, commonly called quantum cryptography, advocates for reliance on the properties of quantum mechanics and thus takes advantage of nature's own fundamental laws in order to achieve security. Today one can easily identify many active areas of research, such as quantum key distribution (see [4; 7; 8; 9] and the more recent [10]), quantum secret sharing [11; 12; 13; 14], quantum teleportation [15; 16], and quantum-secure direction communication [17; 18; 19], to name just a few. Another notable direction that aims to achieve the tenets of privacy and security through quantum mechanics is quantum private comparison (QPC from now on), initiated by [PERSON] and [PERSON] in their pioneering work [20]. [PERSON] and [PERSON] were the first to present a quantum protocol that enables [PERSON] and [PERSON] to find out if their private numbers are equal, without any party gaining additional information in case the numbers are different. This problem can be traced back to the fundamental works [21; 22] by [PERSON], wherein the millionaires' problem was introduced: two millionaires wish to know who is richer, but without divulging any information about the precise amount of their fortune to the other one. Following this line of thought, [PERSON] [23] proposed a classical protocol for the socialist millionaires' problem, where two millionaires desire to find out if they are equally rich. [PERSON] and [PERSON]'s protocol is based on EPR pairs for comparing information with respect to equality and employs the help of a third party (TP). Almost simultaneously, the authors in [24] proposed an alternative QPC protocol utilizing triplets.