## 2 The Proposed Quantum Secret Sharing Protocol and Its Security  
The schematic diagram of the proposed QSS protocol is shown in Figure 1. \(n\) players (\(\text{Bob}_{1}\), \(\text{Bob}_{2}\), \(\ldots\), \(\text{Bob}_{n}\)) are linked with the dealer by a single quantum channel such as a inter-satellite channel (see Section 3.1 for details). For each round of quantum transmission, the first player (\(\text{Bob}_{1}\)) at one far end of the link generates a pair of independent Gaussian random numbers \(\{x_{1},p_{1}\}\) (zero mean) and uses them to modulate the output of the local THz source through amplitude and phase modulators to prepare a GATS \(|x_{1}+jp_{1}\rangle\), where \(j\) denotes the imaginary unit. The state prepared by \(\text{Bob}_{1}\) is then sent to the adjacent player \(\text{Bob}_{2}\). For now, \(\text{Bob}_{2}\) also prepares a GATS \(|x_{2}+jp_{2}\rangle\) and couples it with the transmitted state from \(\text{Bob}_{1}\) into the same spatiotemporal mode through a HABS (the transmittance \(T_{B}\cong 1\)). All the other players in the link perform similar operations. At final, the state that the dealer receives can be expressed as \(\left|\sum_{i=1}^{n}\sqrt{T_{i}}x_{i}+j\sum_{i=1}^{n}\sqrt{T_{i}}p_{i}\right\rangle\), where \(T_{i}\) is the channel transmittance experienced by the quantum signal between the dealer and the \(i\)-th player. The dealer uses the heterodyne detector to measure quadrature components of the received states. In the case of heterodyne detection, the quantum signal is split using a balanced beam splitter. One arm is used to measure the quadrature component \(X\) and the other one is used to measure the quadrature component \(P\) after \(\pi/2\) phase shift of local oscillator. This operation can allow the dealer to share a secure key which can only be recovered by all \(n\) players in cooperation but not by any subset of less than \(n\) players. The details of the protocol are as follows. **Step 1** (Preparation). For each round of quantum transmission, \(n\) players (\(\text{Bob}_{1}\), \(\text{Bob}_{2}\), \(\ldots\), \(\text{Bob}_{n}\)) each locally prepare a thermal Gaussian state \(|x_{i}+jp_{i}\rangle(i=1,2,\ldots,n)\) based on THz sources, Gaussian random numbers \(\{x_{i},p_{i}\}_{i=1}^{n}\) and modulators at their stations. **Step 2** (Transmission). First of all, the first player \(\text{[PERSON]}_{1}\) sends the prepared state \(|x_{1}+jp_{1}\rangle\) to the nearest player \(\text{[PERSON]}_{2}\). After receiving \(\text{[PERSON]}_{1}\)'s quantum state, \(\text{[PERSON]}_{2}\) couples his state and the received state to the same spatiotemporal mode by using the HABS. The merged quantum state is then sent to the next player. The remaining players in the link perform similar operations, so that they can inject the locally prepared state into the same spatiotemporal mode as \(\text{[PERSON]}_{1}\). Finally, the state arriving at the dealer's station can be expressed as \(\left|\sum_{i=1}^{n}\sqrt{T_{i}}x_{i}+j\sum_{i=1}^{n}\sqrt{T_{i}}p_{i}\right\rangle\). **Step 3** (Detection). After receiving the quantum signal state, the dealer performs heterodyne detection to measure its quadrature components and then obtain the measurement results \((x_{r},p_{r})\) which are kept as raw data. Repeat the above procedure many rounds until the dealer obtains sufficient raw data. Figure 1: Structure diagram of THz-CVQSS system. HABS, highly asymmetric beam splitter; HED, heterodyne detector. **Step 4** (Post-processing). The remaining steps use classical post-processing technologies to process these raw data. _Step 4.1._ The dealer and all the players randomly choose and disclose a group of the raw data to deduce the channel transmittances \(\{T_{i}\}_{i=1}^{n}\)[32]. Then all the players discard their disclosed Gaussian random numbers to prevent the eavesdropper from obtaining information about the key. _Step 4.2._ The dealer assume that all the players except [PERSON]\({}_{1}\) are dishonest (if all the players are dishonest, then the QSS protocol is meaningless).