# A Quantum Detectable Byzantine Agreement Protocol Using Only EPR Pairs
## 4 The \(n\) Player EPRQDBA Protocol
### The \(n\) Player Agreement Phase
The crucial assumption (**DBA\({}_{2}\)**) of any detectable Byzantine Agreement protocol, stipulates that all loyal generals either follow the same order or abort the protocol (recall Definition 2). When there is only one loyal general, maybe [PERSON] herself, or one of the lieutenant generals, any decision taken would be fine, as there is no other loyal party that to agree with. ## 5 Discussion and Conclusions

This paper introduced the quantum protocol EPRQDBA that achieves Detectable Byzantine Agreement for an arbitrary number \(n\) of players. The EPRQDBA protocol has a number of advantages to offer. * It is completely general because it can handle any number \(n\) of players. * It takes only a constant number of rounds, namely 3, to complete, no matter how many players there are. * It requires only EPR (specifically \(|\Psi^{*}\rangle\)) pairs, irrespective of the number \(n\) of players. This is probably the most important quality of the protocol. Without a doubt, EPR pairs are the easiest to produce among all maximally entangled states. In comparison, even \(|GHZ_{n}\rangle\) states, would not be able to scale well as \(n\) increases, not to mention other more difficult to create entangled states. All the above qualities advocate for its scalability and its potential applicability in practice. To drive further this point, we present the numerical characteristics of the EPRQDBA protocol in the following Table 1.

Parameter \(m\) is independent of the number of players \(n\), which is another plus for the scalability of the protocol. Obviously, a proper value for must be selected in order to ensure that the probability that a traitor can forge a command vector is negligible. The reason why \(m\) does not scale with \(n\) is that the protocol always uses EPR pairs, that is, bipartite entanglement. The rules of the protocol dictate that every consistency check takes place between two command vectors, or between a command vector and a bit vector. Therefore, in every case the comparison involves just 2 strings of bits, irrespective of the number of generals. Moreover, this comparison, even in the most general case, involves just 2 bits, say \(i\) and \(j\), in every \(n-1\) tuple. Thus, the situation from a probabilistic point of view is always identical with the 3 player case. In a final analysis, the probability that a traitor deceives a loyal general, is the probability of picking the one correct configuration out of many. The total number of configurations is equal to the number of ways to place \(\approx\frac{m}{4}\) identical objects (either 0 or 1, depending on the order) into \(\approx\frac{m}{2}\) distinguishable boxes (the uncertain tuples). The probability that a traitor places _all_ the \(\approx\frac{m}{4}\) bits correctly in the \(\approx\frac{m}{2}\) uncertain tuples is

\[P(\text{ traitor cheats })\approx\frac{1}{\binom{m/2}{m/4}}\, \tag{46}\]

which tends to zero as \(m\) increases. **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. **Institutional Review Board Statement:** Not applicable.