## 2 Theoretical Framework and Simulation Experiment Setup
\(\gamma_{D,n}\) and \(\gamma_{n}\) represent the carrier recombination rates; \(\gamma_{D,21}\) and \(\gamma_{21}\) denote the intradot relaxation rates at which spin-polarized carriers relax from the excited state to the spin-up (down) ground state; \(\gamma_{D,0}\) and \(\gamma_{0}\) are the rates of carrier capture from the WL into the excited state; \(\gamma_{D,s}\) and \(\gamma_{s}\) correspond to the spin relaxation rates; \(\gamma_{D,p}\) and \(\gamma_{p}\) represent the birefringence rates; and \(\gamma_{D,\gamma}\) and \(\gamma_{\gamma}\) are related to the dichroism rates. \(\tau\) indicates the feedback time along any of the delay lines (DL1-DL4) shown in Figure 2; \(\omega_{G}\) is the resonant frequency of the light emitted from the ground state; and \(\omega_{E}\) is the resonant frequency of light emitted from the excited state. \(\Delta\omega_{G}\) represents the frequency detuning between CW\({}_{1}\) (CW\({}_{2}\)) and the ground state emission of the reservoir QD Spin-VCSEL; \(\Delta\omega_{E}\) denotes the frequency detuning between CW\({}_{3}\) (CW\({}_{4}\)) and the excited state emission of the reservoir QD Spin-VCSEL. \(\beta_{sp}\) is the rate of spontaneous emission, also viewed as an indicator of noise strength. The terms \(\xi_{D,GS}^{\pm}\), \(\xi_{D,ES}^{\pm}\), \(\xi_{DS}^{\pm}\) and \(\xi_{ES}^{\pm}\) embody independent Gaussian white noise sources with zero mean and unit variance. \(k_{f}\) is the feedback coupling strength; \(k_{inj}\) stands for the strength of optical injection. \(E_{inj}^{1}\) and \(E_{inj}^{2}\) are the slowly varying complex amplitudes of the CW\({}_{1}\) and CW\({}_{2}\), which are converted to RCP and LCP by the PCL\({}_{2}\) and PCL\({}_{3}\); \(E_{inj}^{3}\) and \(E_{inj}^{4}\) are the injected optical fields for the CW\({}_{3}\) and CW\({}_{4}\), likewise converted by the PCL\({}_{4}\) and PCL\({}_{5}\). \(E_{inj}^{1}\) and \(E_{inj}^{2}\) account for the light fields \(E_{GS}^{+}\) and \(E_{GS}^{-}\), respectively, while \(E_{inj}^{3}\) and \(E_{inj}^{4}\) are charged with generating the optical fields \(E_{ES}^{+}\) and \(E_{GS}^{-}\). The total pump strengths \(\eta=\eta^{+}+\eta^{-}\), \(\eta^{+}\) and \(\eta^{-}\) are the pump intensities for the RCP and LCP components, respectively. The left and right circular polarization components of the GS and ES emissions of the driving QD Spin-VCSEL are replaced with the orthogonal linear components as follows:  
\[ \begin{split} E_{D,Gx}&=\frac{E_{+,GS}^{+}+E_{GS} ^{-}}{\sqrt{2}},\quad E_{D,Gy}&=-i\frac{E_{D,CS}^{+}-E_{D,GS}^{ -}}{\sqrt{2}},\\ E_{D,Ex}&=\frac{E_{D,ES}^{+}+E_{D,ES}^{-}}{\sqrt{2}},\quad E_{D,Ey}&=-i\frac{E_{D,ES}^{+}-E_{D,ES}^{-}}{\sqrt{2}}. \end{split}  \tag{11}\]  
The left and right circular polarization components of the GS and ES emissions of the reservoir QD Spin-VCSEL are rewritten in terms of the orthogonal linear components as follows:  
\[ \begin{split} E_{Gx}&=\frac{E_{GS}^{+}+E_{GS}^{-} }{\sqrt{2}},\quad E_{Gy}&=-i\frac{E_{GS}^{+}-E_{GS}^{-}}{\sqrt{2 }},\\ E_{Ex}&=\frac{E_{ES}^{+}+E_{ES}^{-}}{\sqrt{2}}, \quad E_{Ey}&=-i\frac{E_{ES}^{+}-E_{ES}^{-}}{\sqrt{2}}.\end{split}  \tag{12}\]  
In the QPSKT and m-QAM (m equals 4, 8, 16) presented in Figure 2a, the QPSK and m-QAM signals can be generated through the process of IQ modulation, where baseband signals (\(b^{1}\)-\(b^{4}\)) modulate a continuous light source. Subsequently, these modulated signals are combined via polarization beam combining techniques. The resulting QPSK and m-QAM signals are characterized by  
\[S_{j}(t)=\frac{1}{2}E_{in,j}(t)\left[\sum\limits_{k=1}^{L_{k}}\cos(\varphi_{1k }^{j})+i\sum\limits_{k=1}^{L_{k}}\cos(\varphi_{2k}^{j})\right], \tag{13}\]  
where the subscript \(j\) = 1 denotes QPSK. The subscripts \(j\) = 2, 3, 4 represent 4-QAM, 8-QAM, and 16-QAM, respectively. When \(j\) = 1 and 2, \(L_{k}\) = 4. If \(j\) = 3, \(L_{k}\) = 8, while \(j\) = 4, \(L_{k}\) = 16. The subscript \(k\) indicates the \(k\)th group of bits in the time sequence.