## 2 Quantum Qubit Commitment Scheme  
In this section, we show our protocol. The relativistic geometry of our protocol is chosen to be same as that of Ref. [10], i.e., [PERSON] and [PERSON] agree on a fixed inertial reference frame with coordinates (\(x\), \(y\), \(z\), \(t\)), within which the space-time point \(O\) is the origin and two points \(Q_{0}=(-x\), \(0\), \(0\), \(x\)) and \(Q_{1}=(x\), \(0\), \(0\), \(x)\) are lightlike separated from \(O\) (see Figure 1). [PERSON] and [PERSON] each have agents, separated in their own secure laboratories, adjacent to each of the points \(O\), \(Q_{0}\), and \(Q_{1}\). Before the commitment, [PERSON] and [PERSON] agree on a security parameter \(N\), and a public fixed qubit set \(\{\{|\psi_{k}\rangle_{\chi}=\alpha_{k}|0\rangle_{\chi}+\beta_{k}|1\rangle_{\chi} \}_{k=1}^{d}\), where \(\{|0\rangle_{\chi},|1\rangle_{\chi}\}\) is an orthonormal basis in \(H_{\chi}\) space, \(\alpha_{k}\) and \(\beta_{k}\) are complex weight factors, and \(d\) is the number of the qubits in the set. In our protocol, the qubit set is constrained by two restrictive conditions: (1) any two of the four qubits \(\alpha|0\rangle_{\chi}\pm\beta|1\rangle_{\chi}\) and \(\beta|0\rangle_{\chi}\pm\alpha|1\rangle_{\chi}\) cannot be selected as elements of the qubit set at the same time for arbitrary \(\alpha\) and \(\beta\) parameters; (2) any two elements, \(|\psi_{0}\rangle_{\chi}=\alpha_{0}|0\rangle_{\chi}+\beta_{0}|1\rangle_{\chi}\) and \(|\psi_{1}\rangle_{\chi}=\alpha_{1}|0\rangle_{\chi}+\beta_{1}|1\rangle_{\chi}\), of the qubit set should satisfy \(\chi{<}\psi_{1}|\psi_{0}{>}\chi<\psi_{0}|\psi_{1}{>}\chi<1-\varepsilon\), where \(\varepsilon\) is the probability of the channel errors. Any set satisfying the above two restrictive conditions can be used as a qubit set of our protocol. The second restrictive condition is related to the non-ideal case that there exist channel errors. In the following, we first present our protocol in an idealized form and ignore the second restrictive condition. In the ideal case, we assume perfect state preparations, transmissions, and measurements. We also make idealizations about the relativistic geometry and signaling speed as Ref. [10], supposing that [PERSON] can signal at precisely light speed, all information processing is instantaneous, and the distances from both parties' labs to the relevant points \(O\), \(Q_{0}\), and \(Q_{1}\) are infinitesimal and can be considered to be negligible. **Commit:** In the commit phase, [PERSON] prepares securely a set of qubits \(\{|\Psi_{i}\rangle\}_{i=1}^{N}\) independently randomly chosen from Bell states and keeps them private. \[ \begin{split}|\phi_{1}\rangle_{\text{AB}}&=\tfrac{1 }{\sqrt{2}}(|0\rangle_{\text{A}}|0\rangle_{\text{B}}+|1\rangle_{\text{A}}|1 \rangle_{\text{B}}),|\phi_{2}\rangle_{\text{AB}}&=\tfrac{1}{ \sqrt{2}}(|0\rangle_{\text{A}}|0\rangle_{\text{B}}-|1\rangle_{\text{A}}|1 \rangle_{\text{B}}),\\ |\phi_{3}\rangle_{\text{AB}}&=\tfrac{1}{\sqrt{2}}(|0 \rangle_{\text{A}}|1\rangle_{\text{B}}+|1\rangle_{\text{A}}|0\rangle_{\text{B }}),|\phi_{4}\rangle_{\text{AB}}&=\tfrac{1}{\sqrt{2}}(|0\rangle_{ \text{A}}|1\rangle_{\text{B}}-|1\rangle_{\text{A}}|0\rangle_{\text{B}}),\end{split}  \tag{1}\]  
At point \(O\), [PERSON] sends qubits A (\(|\ \rangle_{\text{A}}\)) of his Bell states to [PERSON] and keeps qubits B (\(|\ \rangle_{\text{B}}\)) to himself. To commit to a qubit \(\alpha|0\rangle_{\chi}+\beta|1\rangle_{\chi}\) in \(H_{\chi}\) space, [PERSON] carries out teleportation via measuring qubits X (\(|\ \rangle_{\text{X}}\)) and A collectively in the Bell basis resulting qubits B on [PERSON]'s side collapse to \(\alpha|0\rangle_{\text{B}}\pm\beta|1\rangle_{\text{B}}\) and \(\beta|0\rangle_{\text{B}}\pm\alpha|1\rangle_{\text{B}}\) with equal probability, and sends over secure channels to her agents at \(Q_{0}\) and \(Q_{1}\) the outcomes of the collective measurement of qubits X and A.