# A Quantum Detectable Byzantine Agreement Protocol Using Only EPR Pairs
## 4 The \(n\) Player EPRQDBA Protocol
### The \(n\) Player Agreement Phase
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Qubits**} \\ \hline \multicolumn{2}{c}{\# of \(|\Psi^{*}\rangle\) pairs} & \multicolumn{2}{c}{\# of \(|\ast\rangle\) qubits} \\ \hline \multicolumn{2}{c}{\((n-1)m\)} & \multicolumn{2}{c}{\((n-2)(n-1)m\)} \\ \hline \multicolumn{2}{c}{Bits \& Messages} \\ \hline Round & \# of messages & \# bits per message & Total \# of bits \\ \hline
1 & \(n-1\) & \((n-1)m\) & \((n-1)^{2}m\) \\ \hline
2 & \((n-2)(n-1)\) & \((n-1)m\) & \((n-2)(n-1)^{2}m\) \\ \hline
3 & \((n-2)(n-1)\) & \(0\leq\sharp\) bpm \(\leq 2(n-1)m\) & \(0\leq\sharp\) b \(\leq\) \\  & & & \(2(n-2)(n-1)^{2}m\) \\ \hline
4 & 0 & 0 & 0 \\ \hline \multicolumn{2}{c}{_m \& Probability_} \\ \hline \(m\) & \(\frac{m}{4}\) & \(\frac{m}{2}\) & \(P(\text{ traitor cheats })\) \\ \hline
4 & 1 & 2 & 0.500000000000000 \\ \hline
8 & 2 & 4 & 0.167 \\ \hline
16 & 4 & 8 & 0.0145 \\ \hline
32 & 8 & 16 & 7.770 \(\cdot 10^{-5}\) \\ \hline
64 & 16 & 32 & 1.663 \(\cdot 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: This table summarizes the numerical characteristics of the EPRQDBA protocol. The abbreviations “bpm” and “tb” stand for “bits per message” and “total bits,” respectively. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable, the study does not report any data. **Conflicts of Interest:** The authors declare no conflict of interest. ## Appendix A Proofs of the Main Results

In this Appendix we formally prove the main results of this paper. **Proposition A1** (Loyal [PERSON]). : _If [PERSON] is loyal, the \(3\) player EPRQDBA protocol will enable the loyal lieutenant general to agree with [PERSON]. Specifically, if [PERSON] and [PERSON] are loyal and [PERSON] is a traitor, [PERSON] will follow [PERSON]'s order. Symmetrically, if [PERSON] and [PERSON] are loyal and [PERSON] is a traitor, [PERSON] will follow [PERSON]'s order._

Proof. : We assume that [PERSON] is loyal and, without loss of generality, that [PERSON] conveys the command \(0\) to both [PERSON] and [PERSON], along with the appropriate command vectors as proof. These command vectors satisfy the properties asserted in Lemma 2. Let us also assume that [PERSON] is loyal. Consequently, [PERSON] will use the Algorithm 1 to verify the validity of the received command vector \(0_{B}\). According to the rules of the \(3\) player EPRQDBA Protocol 1, the only way for [PERSON] to change his initial decision and abort is if [PERSON] sends to [PERSON] a valid command vector \(\mathds{1}_{C}\). In such a case [PERSON] would be a traitor that has in reality received the command vector \(0_{C}\) during \((\mathbf{Round}_{1})\). Can [PERSON] construct a convincing \(\mathds{1}_{C}\)? [PERSON], having received the command vector \(0_{B}\), knows the positions of all the pairs in a of the form \(a_{2k+1}a_{2k}\), such that \(a_{2k+1}=0\) and \(a_{2k}=1,0\leq k\leq m-1\). \[\mathbb{P}_{0,1}(0_{B})=\{p_{1},p_{2},\ldots,p_{t}\}\,\ \text{where}\ t\approx\frac{m}{4}. \tag{1}\]

Having \(0_{C}\), [PERSON] knows that the least significant bit of the \(\approx\frac{m}{2}\) uncertain pairs in \(0_{C}\) is \(1\) and the most significant bit is \(0\) or \(1\), with equal probability \(0.5\). However, [PERSON] cannot know with certainty if the most significant bit of a specific uncertain pair is \(0\) or \(1\). Therefore, when constructing \(\mathds{1}_{C}\), [PERSON] can make two detectable mistakes. \((\mathbf{M}_{1})\) Place a \(0\) in a wrong pair that is not actually contained in \(\mathbb{P}_{0,1}(0_{B})\). \((\mathbf{M}_{2})\) Place a \(1\) in a wrong pair that appears in \(\mathbb{P}_{0,1}(0_{B})\). The situation from a probabilistic point of view resembles 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 \((0)\) into \(\approx\frac{m}{2}\) distinguishable boxes.