  Rational Identity Testing (RIT) is the decision problem of determining
whether or not a noncommutative rational formula computes zero in the free skew
field. It admits a deterministic polynomial-time white-box algorithm [Garg,
Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018);
Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen
and Makam (2017)] in the black-box setting, via singularity testing of linear
matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in
the white-box setting follows from the result of Derksen and Makam (2017).
  Designing an efficient deterministic black-box algorithm for RIT and
understanding the parallel complexity of RIT are major open problems in this
area. Despite being open since the work of Garg, Gurvits, Oliveira, and
Wigderson (2016), these questions have seen limited progress. In fact, the only
known result in this direction is the construction of a quasipolynomial-size
hitting set for rational formulas of only inversion height two [Arvind,
Chatterjee, Mukhopadhyay (2022)].
  In this paper, we significantly improve the black-box complexity of this
problem and obtain the first quasipolynomial-size hitting set for all rational
formulas of polynomial size. Our construction also yields the first
deterministic quasi-NC upper bound for RIT in the white-box setting.
