## 3 Results and Discussion
### Noise Characterization  
In the conditions previously described, a Quantum-to-classical Noise Ratio (QCNR) of at least 11.7 dB was always observed, which highlights the high noise clearance obtained. Over \(1\times 10^{9}\)  samples, a classical noise floor of \(1.04\times 10^{-6}\,\mathrm{V}^{2}\) was evaluated, while an average variance of \(1.64\times 10^{-5}\,\mathrm{V}^{2}\) was seen for the homodyne signal. As can be seen in Figure 2a, both noise sources follow the expected null-mean Gaussian distribution. After applying the binning algorithm, as can be seen in Figure 2b, these output signals appear to be successfully mapped to the same set of output codes. Nonetheless, both sets seem to significantly deviate from the theoretically expected uniform distribution, whose probability density function is represented by the dashed line. This is additionally corroborated by the fact that both datasets conclusively fail a Chi-square goodness-of-fit test with null \(p\)-values, thus rejecting the null hypothesis that the measurements observed arise from a uniform distribution. Consequently, we verify that the simple binning algorithm is not completely successful in suppressing all the underlying biases of the raw RNG output. This illustrates the importance of implementing the RE layer in a fully fledged practical application. Despite this, the CS seems to present a greater deviation than the raw QRNG output, which may be a consequence of the increased entropy introduced by the QS. In Figure 3a,b, the normalized autocorrelation coefficients for 10 M measurements calculated over a delay of 200 samples can be seen, respectively, for the raw and uniformized dataset. For a sequence of this size, these values should be normally distributed around a null average value with a standard deviation of \(3.16\times 10^{-4}\), which is here represented by a black-dashed line. Although the coefficients for the uniformized output stay within the expected standard deviation in both scenarios, this clearly contrasts with the case seen for the raw sequences presented in Figure 3a. Here, the QS clearly shows significant correlations for delays up to 60  samples, which can be mostly attributed to the TIA [36]. Meanwhile, we confirm the highly correlated nature of the CS output, which clearly shows the unreliability of using electronic noise for randomness generation. Consequently, we expect the raw CS to perform significantly worse at prime number generation. Although the RE layer was not implemented, a quantum min-entropy of 8.39 bit was calculated for the QRNG implementation through the evaluation of Equation (3). At the chosen sampling rate, an information-theoretically secure implementation would support a maximum output rate of approximately 8.25 Gbps. This high throughput allows the implementation of a fast prime generation scheme. For example, as can be assessed through Equation (4), a theoretical prime generation rate of 180.0 M primes per second could be expected for 32-bit primes. Naturally, this does not consider the algorithmic complexity of the prime-testing layer, and significantly lower rates would be seen in a real-time implementation. Moreover, as expected, this rate decreases for prime numbers with longer bit-lengths since, according to the prime number theorem, the density of primes for any given value \(x\) is asymptotically \(\frac{1}{log(x)}\)[4]. Consequently, the probability of the RNG outputting a prime number decreases for larger bit sizes. For example, for a length of 512 bit, the expected output rate is 11.6 M primes per second. Figure 3: Autocorrelation coefficients for 10 M classical and homodyne noise samples, calculated before (**a**) and after (**b**) the binning algorithm was applied. The black-dashed line represents the theoretical standard deviation expected. Figure 2: Distribution of 100 M samples of homodyne noise and classical noise taken (**a**) before and (**b**) after the binning algorithm was applied. This equiprobable binning maps both Gaussian distributions to the same output space.