# A Quantum Detectable Byzantine Agreement Protocol Using Only EPR Pairs
## 4 The \(n\) Player EPRQDBA Protocol
### The \(n\) Player Agreement Phase
Without loss of generality, we may suppose that the order in question is 0. Then, at the end of (**Round\({}_{2}\)**), each loyal LT\({}_{i}\) communicates to every other general his initial decision 0 and proof \(\theta_{i}\). Let us examine the possible actions of the disloyal lieutenant generals at the end of (**Round\({}_{2}\)**). \(\Box\) At least one disloyal LT\({}_{k}\) transmits to at least one loyal LT\({}_{i}\) his initial decision 1 with a consistent proof \(\mathds{1}_{k}\). Then, at the end of (**Round\({}_{3}\)**), LT\({}_{i}\) will communicate to every other lieutenant general his intermediary decision \(d_{i}^{(3)}=\bot\), together with his proof \(\mathds{1}_{k}\). This will cause all other loyal lieutenant generals to set their final decision to \(\bot\) during (**Round\({}_{4}\)**), thereby achieving agreement by aborting at the end of the protocol. \(\Box\) Every disloyal LT\({}_{k}\) tells each loyal LT\({}_{i}\) that his initial decision is either \(\bot\), or 0, with proof \(\theta_{k}\). Then, at the end of (**Round\({}_{3}\)**), every LT\({}_{i}\) will communicate to every other general his intermediary decision 0, together with his intermediary proof \(\rho_{i}^{(3)}=\varepsilon\). In this scenario, all loyal lieutenant generals set their final decision to 0 during (**Round\({}_{4}\)**), thereby agreeing to follow order 0 at the end of the protocol. \(\bullet\) During (**Round\({}_{2}\)**) at least one loyal LT\({}_{i}\) receives the order 0 and the consistent command vector \(\mathds{1}_{j}\). In this scenario, it does not matter what the remaining disloyal lieutenant generals do at the end of (**Round\({}_{2}\)**). Accordingly, at the end of (**Round\({}_{2}\)**), LT\({}_{i}\) communicates to every other lieutenant general his initial decision 0 and proof \(\theta_{i}\), and LT\({}_{j}\) communicates to every other lieutenant general his initial decision 1 and proof \(\mathds{1}_{j}\). Then, during (**Round\({}_{3}\)**), all loyal lieutenant generals set their intermediary decision to \(\bot\), and, during (**Round\({}_{4}\)**), all loyal lieutenant generals set their final decision to \(\bot\), thereby achieving agreement by aborting at the end of the protocol. \(\bullet\) During (**Round\({}_{2}\)**) some loyal lieutenant generals receive and verify the same order from [PERSON], which, without loss of generality, we may suppose that is 0, and the remaining loyal lieutenant generals receive inconsistent command vectors from [PERSON]. By employing the arguments used in the previous case, it is straightforward to conclude that all loyal lieutenant generals reach agreement in any eventuality. \(\Box\) Ergo, all loyal lieutenant generals will reach agreement. \(\Box\)

## References

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