## 3 Results and Discussion
### Length-Agnostic Statistical Validation
As can be seen, no persisting correlations were found for either source. The same is seen in the correlation analyses for all other lengths, which can be found in Figure 7 from Appendix A. Additionally, the proportion of measurements that fall below the equiprobable threshold was calculated for increasing lengths, as shown in Figure 7b. Here, the theoretically expected Confidence Interval (CI), given by Equation (8), are represented by full lines. Nonetheless, since only 10 M samples were evaluated, its 99% binomial CI was also calculated for each of the probability bounds. As can be seen, all the obtained values appear outside the acceptable interval. Consequently, we are forced to conclude that the prime RNG fails this particular statistical test for all the cases that were considered. Given that a single assessment was made, additional test runs are necessary to assess the statistical significance of this result. Nonetheless, this indicates an unbalanced dataset where the proportions of 0's and 1's in the binary sequences are significantly different. In practice, this means that the prime generation of the implemented RNG is slightly unbalanced, with more values being consistently outputted below the calculated threshold even for the larger prime lengths. Finally, to conclusively determine the significance of this result, a 100 Mbit binary sequence obtained from the 32-bit prime output was submitted to NIST's statistical test  
Figure 7: **(a)** Normalized autocorrelation coefficients for 10 M threshold values for the 32-bit uniformized sources. The black-dashed line represents the theoretical standard deviation expected. (b) Proportion of the 10 M samples that fall below the equiprobable threshold. Full lines represent the theoretical maximum expected interval, while black-dashed lines represent its 99% CI. suit [1]. These results are represented in Table A1 from Appendix B, both for the uniformized CS and QS. Here, each binary sequence was subdivided into bitstreams of 1 Mbit and subsequently submitted to the statistical tests. As can be seen, the output conclusively fails most of the applied evaluations, which confirms our prior assessment. In fact, we verify that both sources decisively fail the _Frequency_ test, which evaluates a very similar metric to the previous analysis. Notably, although it also yields a failure, the QS performed slightly better in the _OverlappingTemplate_ and _NonoverlappingTemplate_ tests. In fact, here, 83 and 73 sequences, respectively, successfully passed these tests, which is contrasted with the 57 and 31 sequences that yielded a positive result for the CS. As described in [1], these two tests assess the occurrence of predetermined bit patterns, which could indicate that the QS output has higher entropy. Moreover, although the QS ultimately fails some test runs and, consequently, is rejected by the overall test, it is relevant to highlight that the vacuum-based QRNG performed significantly better than the classical scheme when considering all test iterations applied. In fact, the QS successfully passed 140 tests of the 161 applied, yielding a success rate of 86.96%. Meanwhile, the CS only passed 45.34% of the tests considered by the statistical suite. Despite this, the failures observed are very significant since they highlight serious flaws in the prime RNG implemented. Although it is hard to give a conclusive reason for this result, a strong candidate is the aforementioned lack of RE layer. The temporal correlations introduced by the TIA and the classical noise floor clearly manifest themselves on the prime number sequences obtained since, as previously verified, the prime-searching algorithm is unable to suppress them. In this way, it would be important to reassess these figures after implementing this essential step. Moreover, additional data should be gathered to allow more extensive statistical validation. This would allow us to conclusively determine the viability of the implemented vacuum-based QRNG for prime number generation.