## 2 The GMCS-Involved \((t,n)\) Scheme for CVQSS  
Enlightened by the characteristics of the GMCS-involved quantum key distribution (QKD) [19] and the topological structure of the \((n,n)\)-threshold scheme using weak coherent states [24], we propose an improved \((t,n)\) threshold scheme for the practical CVQSS. As shown in Figure 1, all participants and the dealer are linked by a single fibre-based quantum channel for transmission. The first participant \(B_{1}\) generates Gaussian random numbers and modulates actively the output of a local laser using phase and amplitude modulators to prepare a coherent state \(|x_{1}+ip_{1}\rangle\). Here, two variables \(x_{1}\) and \(p_{1}\) are independent Gaussian random numbers with zero mean and a variance of \(V_{1}N_{0}\), where \(N_{0}\) is the shot-noise variance, and \(V_{1}\) represents the modulation variance determined by \(B_{1}\). Then, the coherent state \(|x_{1}+ip_{1}\rangle\) is sent to the adjacent participant and passes through a highly asymmetric beam splitter of \(B_{2}\). Meanwhile, \(B_{2}\) prepares the other GMCS and couples it into the spatiotemporal mode as the same as \(B_{1}\) by using the second beam splitter. \(B_{2}\) could realize phase-space displacements of \(\{x_{2},p_{2}\}\) by adjusting the modulation variances and the reflectivity of the asymmetric beam splitter. The remaining participants execute the similar procedure. Finally, the quantum state of dealer can be expressed as \(|\sum_{j=1}^{n}\sqrt{T_{j}}x_{j}+i\sum_{j=1}^{n}\sqrt{T_{j}}p_{j}\rangle\), where \(T_{j}\) is the channel transmittance from the \(jth\) participant to the dealer. The dealer achieves \(\{x_{h},p_{h}\}\) by measuring the amplitude and phase quadratures of the end quantum state via double homodyne detection. Stage 1. Preparation Processing:  
1. The first participant \(B_{1}\) generates the coherent state \(|x_{1}+ip_{1}\rangle\) based on its Gaussian random numbers \(\{x_{1},p_{1}\}\), laser and modulator, and sends it to the next participant \(B_{2}\). 2. Each of the remained participants couples the locally prepared GMCS into the spatiotemporal mode at the same time as \(B_{1}\) by using the highly asymmetric beam splitter. 3. The dealer obtains \(\{x_{h},p_{h}\}\) by measuring the amplitude and phase quadratures of the received quantum state via double homodyne detection. The resulting \(\{x_{h},p_{h}\}\) is saved as raw data. 4. Repeat above steps until the dealer gets enough raw data. 5. A subset of the raw data is randomly chose and the dealer demands all the participants to publish the corresponding Gaussian random numbers. Combined with the corresponding measurement results, the transmittance \(\{T_{1},T_{2},\cdots,T_{n}\}\) can be achieved [32]. All participants abandon the disclosed data. 6. The dealer randomly chooses a subset of remaining raw data after step 5. The dealer presumes each participant except \(B_{m}\) is dishonest and demands them to publish their corresponding Gaussian random numbers. 7. The dealer replaces the measurement result by \(x_{F}=x_{h}-\sum_{j=1,j\
eq m}^{n}\sqrt{T_{j}}x_{j}\) and \(p_{F}=p_{h}-\sum_{j=1,j\
eq m}^{n}\sqrt{T_{j}}p_{j}\). In this case, \(\{x_{F},p_{F}\}\) and the raw data of \(B_{m}\) are same subsets. Therefore, the dealer and \(B_{m}\) can gain a lower bound of secure key rate \(R_{m}\) of the GMCS-based quantum cryptography. 8. Repeat the step 7 for \(n\) times. Finally, the dealer gets secure key rates \(\{R_{1},R_{2},\cdots,R_{n}\}\)[18]. Stage 2. Implementation Processing:  
1. Legitimate \(t\) participants are sorted by an agreed rule using \(C_{lj}\) for \(l\in\{1,2,\cdots,t\}\) and \(j\in\{1,2,\cdots,n\}\). 2. Supposing that the sequence of \(t\) participants is shown in Figure 2, the dealer randomly chooses a subset of the remained raw data and demands all participants except \(C_{12}\) to publish the corresponding Gaussian random numbers.