## 2 Results
### The Relationship between the Two Protocols  
Now, we show that, in principle, the above experimental protocol is a faithful implementation of the theoretical one. In the experimental protocol, following the occupation number representation widely used in quantum optics [57], at time \(t\) if there is a wave packet of a photon on path \(X\) and no wave packet on path \(Y\), the state can be denoted by \(|1\rangle_{X}|0\rangle_{Y}\). Conversely, if there is a wave packet on path \(Y\) and no wave packet on path \(X\), the state can be denoted by \(|0\rangle_{X}|1\rangle_{Y}\). To make the time \(t\) more explicit, let us write them as \(|t\rangle_{X}|0\rangle_{Y}\) and \(|0\rangle_{X}|t\rangle_{Y}\), respectively. That is, we use the symbol \(t\) in \(|\ldots\rangle\) to denote the time that the wave packet of a single photon presents in the path, instead of the number of photons, and \(|0\rangle\) to denote that no wave packet is presented in the path at any time. Obviously, the state \(|t\rangle_{P}\) is orthogonal to \(|t^{\prime}\rangle_{P}\) (\(P=X,Y\)) for any \(t\
eq t^{\prime}\) and they are all orthogonal to \(|0\rangle_{P}\). For simplicity, suppose that, except for \(SR_{A}\), \(SR_{B}\), \(SR_{x}\), and \(SR_{y}\), the time for the photon to travel through all other devices in Figure 1 is negligible. Under this formalism, when [PERSON] sends the photon \(\Psi_{j}\) (\(j=1,\ldots,s\)) from the source \(S_{b}\) at time \(t_{j}\), the initial state of \(\Psi_{j}\) after passing \(BS_{A}\) is as shown:  
\[\big{|}\Psi_{j}\big{\rangle}_{ini}=\frac{1}{\sqrt{2}}(\big{|}t_{j}\big{\rangle} _{X}|0\rangle_{Y}+(-1)^{b}|0\rangle_{X}\big{|}t_{j}\big{\rangle}_{Y}). \tag{14}\]  
After passing \(SR_{A}\), which introduces the delay time \(\tau_{j}\) to path \(Y\), the state of \(\Psi_{j}\) that left [PERSON]'s site is as below:  
\[\big{|}\psi_{j}\big{\rangle}_{A}=\frac{1}{\sqrt{2}}(\big{|}t_{j}\big{\rangle} _{X}|0\rangle_{Y}+(-1)^{b}|0\rangle_{X}\big{|}t_{j}+\tau_{j}\big{\rangle}_{Y}). \tag{15}\]  
In the unveil phase, when [PERSON] learns [PERSON]'s delay time \(\tau_{j}\) and sets \(SR_{B}\) accordingly, the final state of the photon \(\Psi_{j}\) arriving at \(BS_{B}\) after passing \(SR_{x}\), \(SR_{y}\) and \(SR_{B}\) is as follows:  
\[\big{|}\psi_{j}\big{\rangle}_{fin} = \frac{1}{\sqrt{2}}(\big{|}t_{j}+\tau_{hold}+\tau_{j}\big{\rangle} _{X}|0\rangle_{Y} \tag{16}\] \[+(-1)^{b}|0\rangle_{X}\big{|}t_{j}+\tau_{hold}\big{\rangle}_{Y})\] \[= \frac{1}{\sqrt{2}}(\big{|}t_{j}^{\prime}\big{\rangle}_{X}|0 \rangle_{Y}+(-1)^{b}|0\rangle_{X}\big{|}t_{j}^{\prime}\big{\rangle}_{Y}),\]  
where \(\tau_{hold}\) is the length of the time that \(\Psi_{j}\) was stored in \(SR_{x}\) and \(SR_{y}\), and  
\[t_{j}^{\prime}\equiv t_{j}+\tau_{hold}+\tau_{j}. \tag{17}\]  
Meanwhile, when combined with \(BS_{B}\), the detectors \(D_{0}\) and \(D_{1}\) serve as the projective operators:  
\[P_{0}\equiv|\psi\rangle_{0}\langle\psi|_{0} \tag{18}\]  
and  
\[P_{1}\equiv|\psi\rangle_{1}\langle\psi|_{1}, \tag{19}\]  
respectively, where  
\[|\psi\rangle_{0}\equiv\frac{1}{\sqrt{2}}(\big{|}t_{B}\big{\rangle}_{X}|0 \rangle_{Y}+|0\rangle_{X}|t_{B}\rangle_{Y}) \tag{20}\]  
and  
\[|\psi\rangle_{1}\equiv\frac{1}{\sqrt{2}}(\big{|}t_{B}\big{\rangle}_{X}|0 \rangle_{Y}-|0\rangle_{X}|t_{B}\rangle_{Y}), \tag{21}\]  
with \(t_{B}\) denoting the time that [PERSON] applies the measurement. Therefore, if [PERSON] takes \(t_{B}=t_{j}^{\prime}\), then, in the ideal case where the error rate \(\varepsilon\) is negligible, the detector \(D_{b}\) should click with certainty where \(b\) is [PERSON]'s committed bit. Otherwise, he knows that [PERSON] cheated. To see that the above presentation of the states is equivalent to that in our theoretical QBC protocol, let us view the time range \([0,\tau_{\max}]\) (within which [PERSON] picks her delay time\(\tau_{j}\)) as a series of time slots \(T_{1}\), \(T_{2}\),..., \(T_{i}\),..., \(T_{n-1}\). Here, \(0\leq T_{i}\leq\tau_{\max}\) (\(i=1,\ldots,n-1\)), and \(T_{i}\
eq T_{i^{\prime}}\) for any \(i\
eq i^{\prime}\).