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We show that the space-bounded Statistical Zero Knowledge classes SZK_L and NISZK_L are surprisingly robust, in that the power of the verifier and simulator can be strengthened or weakened without affecting the resulting class. Coupled with other recent characterizations of these classes [Eric Allender et al., 2023], this can be viewed as lending support to the conjecture that these classes may coincide with the non-space-bounded classes SZK and NISZK, respectively. 

Abstract Primordial germ cells (PGCs) are the germline precursors that give rise to oocytes and sperm, ensuring the continuation of life. While the PGC specification is extensively studied, it remains elusive how the PGC population is sustained and expanded after they migrate to embryonic gonads before birth. This study demonstrates that NRF1, a known regulator for mitochondrial metabolism, plays critical roles in post‐migrating PGC development. We show that NRF1 protein level gradually increases in post‐migrating PGCs during embryonic development. ConditionalNrf1knockout from embryonic germ cells leads to impaired PGC proliferation and survival. In addition, NRF1 may also actively drive PGC derivation from pluripotent stem cells. Using whole genome transcriptome profiling and ChIP‐seq analyses, we further reveal that NRF1 directly regulates key signalling molecules in PGC formation, transcription factors in proliferation and cell cycle and enzymes in mitochondrial metabolism. Overall, our findings highlight an essential requirement of NRF1 in regulating a broad transcriptional network to support post‐migrating PGC development both in vitro and in vivo.