## 4 Conclusions  
In conclusion, we have implemented a probabilistic prime quantum number generation scheme based on homodyne measurements of vacuum fluctuations passed through a set of 24  Miller-Rabin primality tests. For a prime length of 512 bits, we expect a theoretical maximum throughput of 11.6 M primes per second with a probability of outputting a composite number of only \(3.55\times 10^{-15}\). By removing the QS, a second classical generation scheme based on electronic noise was obtained, and the two schemes were compared. Although the overall number of primes does not depend on the noise source chosen, the quantum scheme clearly outperforms the CS at small prime generation. For the set of 32-bit primes, increases in the diversity of primes up to 585.0% are shown, which cannot be explained by the bias of the input distribution. In fact, the quantum source was shown to yield approximately 5% of all possible prime outcomes, outperforming the classical RNG by an order of magnitude. Moreover, we revealed that strong biases in the input distribution are clearly reflected in the prime output distribution, and although these differences are less prominent for larger lengths, the output distributions were also shown to consistently improve for the QS. Finally, a novel length-agnostic statistical test for prime number sequences based on establishing interval bounds for the expected frequency of measurements in an equiprobable binning of the prime distribution was proposed and applied to the output of the uniformized sources. Using the known bounds for the prime-counting function, the proportion of prime numbers below any given threshold can be bounded such that observations outside this range are witnesses of an unbalanced prime distribution. This approach allows the direct validation of a sequence of prime numbers with arbitrary length instead of relying on assessing the underlying RNG, which would not consider potential biases introduced by the prime-searching algorithm. This analysis was able to reveal the underlying correlations in the prime output of both sources, which remain present due to the lack of a RE layer. The resultant binary sequences thus conclusively fail the tests contained in the NIST statistical test suite, with only 86.96% and 45.34% of the evaluations being, respectively, passed by the QS and CS. In fact, out of the 161 evaluations applied, the QS source passed 140 tests, whereas the CS had a pass rate of 73. This nevertheless shows a significant difference between the two cases. Finally, to improve the statistical significance of this analysis, it would be important to gather additional datasets for each of the cases considered. This would allow us to more precisely estimate the key merit figures by establishing their respective confidence intervals, as well as apply a more extensive statistical validation. Moreover, to conclusively validate the viability of the vacuum-based implementation, it is fundamental to reassess all these cases after implementing the required RE stage. Conceptualization, M.J.F., N.J.M. and N.A.S. ; Methodology, M.J.F., N.J.M. and N.A.S. ; Software, M.J.F. ; Validation, M.J.F. ; Formal Analysis, M.J.F. ; Investigation, M.J.F., N.J.M. and N.A.S. ; Resources, N.J.M. and N.A.S. ; Writing--Original Draft Preparation, M.J.F. ; Writing--Review and Editing, M.J.F., N.J.M., N.A.S. and A.N.P. ; Visualization, M.J.F. ; Supervision, M.J.F., N.J.M., N.A.S. and A.N.P. All authors have read and agreed to the published version of the manuscript. This work was supported in part by Fundacao para a Ciencia e a Tecnologia (FCT) through national funds, by the European Regional Development Fund (FEDER), through the Competitiveness and Internationalization Operational Programme (COMPETE 2020) of the Portugal 2020 framework, under the project QuantumPrime (PTDC/EEI-TEL/8017/2020) and UIDB/50008/2020-UIDP/50008/2020. [PERSON] also acknowledges the 2022.09584.BD PhD grant from FCT. Not applicable. The datasets presented in this study are available on request from the corresponding author.