We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $BPTIME$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows.
1. A new pseudorandom generator and its consequences: We build on techniques developed to prove hierarchy theorems for probabilistic time with advice (Fortnow and Santhanam, FOCS 2004) to construct the first unconditional pseudorandom generator of polynomial stretch computable in pseudodeterministic polynomial time (with one bit of advice) that is secure infinitely often against polynomial-time computations. As an application of this construction, we obtain new results about the complexity of generating and representing prime numbers. For instance, we show unconditionally for each $\varepsilon > 0$ that infinitely many primes $p_n$ have a succinct representation in the following sense: there is a fixed probabilistic polynomial time algorithm that generates $p_n$ with high probability from its succinct representation of size $O(|p_n|^{\varepsilon})$. This offers an exponential improvement over the running time of previous results, and shows that infinitely many primes have succinct and efficient representations.
2. Structural results for probabilistic time from pseudodeterministic algorithms: Oliveira and Santhanam (STOC 2017) established unconditionally that there is a pseudodeterministic algorithm for the Circuit Acceptance Probability Problem ($CAPP$) that runs in sub-exponential time and is correct with high probability over any samplable distribution on circuits on infinitely many input lengths. We show that improving this running time or obtaining a result that holds for every large input length would imply new time hierarchy theorems for probabilistic time. In addition, we prove that a worst-case polynomial-time pseudodeterministic algorithm for $CAPP$ would imply that $BPP$ has complete problems.
3. Equivalence between pseudodeterministic constructions and hierarchies: We establish an equivalence between a certain explicit pseudodeterministic construction problem and the existence of strong hierarchy theorems for probabilistic time. More precisely, we show that pseudodeterministically constructing in exponential time strings of large $rKt$ complexity (Oliveira, ICALP 2019) is possible if and only if for every constructive function $T(n) \leq \exp(o(\exp(n)))$ we have $BPTIME[poly(T)] \not\subseteq i.o.BPTIME[T]/\log T$.
More generally, these results suggest new approaches for designing pseudodeterministic algorithms for search problems and for unveiling the structure of probabilistic time.
