© 2018, Duke University Press. Let Ω⊂ Rn+1, n≥ 1, be a corkscrew domain with Ahlfors-David regular boundary. In this article we prove that ∂Ω is uniformly n-rectifiable if every bounded harmonic function on Ω is ε-approximable or if every bounded harmonic function on Ω satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when Ω = Rn+1 \ E and E is Ahlfors-David regular. Our results establish a conjecture posed by Hofmann, Martell, and Mayboroda, in which they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability, one in terms of the so-called S < N estimates and another in terms of a suitable corona decomposition involving harmonic measure.