## 7 Discussion and Conclusions  
This paper presents the innovative GHZ\({}_{3}\)MQPEC protocol that accomplishes a multi-party quantum private comparison leveraging maximally entangled \(|GHZ_{3}\rangle\) triplets. We list below some of the most important advantages of the protocol.  
* The primary aim of this protocol is to be easily executable by modern quantum computers. This is achieved by employing only \(|GHZ_{3}\rangle\) triplets, regardless of the number of participants. Though high-dimensional quantum states and more intricate multiparticle entangled states are feasible, their creation is challenging given the current state of quantum technology, leading to longer preparation times, particularly in situations involving many players. By exclusively using \(|GHZ_{3}\rangle\) states, which are simpler to produce than Bell states, these issues are mitigated, advancing the practical implementation of the protocol.
* A key quantitative feature of this protocol is that the number of players and the volume of data being compared determine the required quantum resources, which scale linearly, further demonstrating its practicality. Notably, the GHZ\({}_{3}\)MQPEC protocol can be executed in both parallel and sequential modes. While the quantum operations are intended to run in parallel, sequential execution is also possible if quantum resources are insufficient, allowing the participants to be divided into smaller groups for sequential processing.
* Additionally, the protocol does not depend on a quantum signature scheme or pre-shared keys, which simplify and reduce its cost. Implementation is further streamlined as all participants use identical private quantum circuits consisting of Hadamard and CNOT gates, making it straightforward to implement on contemporary quantum computers.
* Crucially, the protocol ensures that external parties cannot access any information about the participants' values, and participants themselves cannot learn each other's secret numbers. The protocol is demonstrated to be information-theoretically secure.  
In closing, we briefly mention the special case \(n=2\), which is when there are only two millionaires, say [PERSON] and [PERSON]. As we mentioned in Section 3, in such a case Sophia becomes unnecessary and [PERSON] alone suffices. The GHZ\({}_{3}\)MQPEC protocol can be easily simplified to function without the presence of Sophia, using the quantum circuit shown in Figure 8. Obviously, the whole mathematical description of the protocol, its quantitative characteristics, and its security remain unaffected.  
The current state of affairs, regarding the availability of quantum computers with a sufficient number of qubits, undoubtedly imposes certain limitations on the practical execution of any quantum protocol, including, of course, GHZ\({}_{3}\)MQPEC. Moreover, it should be noted that measuring \(m\) triplets simultaneously can be challenging and may call for sophisticated multichannel detection systems, which raises the complexity of the implementation. Hopefully, the upcoming generation of quantum computers will address many of these practical issues and allow for increased scalability.  
**Author Contributions:** Conceptualization, T.A. and A.S.; methodology, T.A.; validation, A.S.; formal analysis, A.S.; investigation, T.A.; writing--original draft preparation, T.A. and A.S.; writing--review and editing, T.A. and A.S.; visualization, A.S.; supervision, T.A.; project administration, T.A. and A.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.