  Equivalence testing, a fundamental problem in the field of distribution
testing, seeks to infer if two unknown distributions on $[n]$ are the same or
far apart in the total variation distance. Conditional sampling has emerged as
a powerful query model and has been investigated by theoreticians and
practitioners alike, leading to the design of optimal algorithms albeit in a
sequential setting (also referred to as adaptive tester). Given the profound
impact of parallel computing over the past decades, there has been a strong
desire to design algorithms that enable high parallelization. Despite
significant algorithmic advancements over the last decade, parallelizable
techniques (also termed non-adaptive testers) have $\tilde{O}(\log^{12}n)$
query complexity, a prohibitively large complexity to be of practical usage.
Therefore, the primary challenge is whether it is possible to design algorithms
that enable high parallelization while achieving efficient query complexity.
  Our work provides an affirmative answer to the aforementioned challenge: we
present a highly parallelizable tester with a query complexity of
$\tilde{O}(\log n)$, achieved through a single round of adaptivity, marking a
significant stride towards harmonizing parallelizability and efficiency in
equivalence testing.
