This paper presents some new results on the computational complexity of the set of uniquely satisfiable Boolean formulas (USAT). Valiant and Vazirani showed that USAT is complete for the class $D^{P}$ under randomized reductions. In spite of the fact that the probability bound of this reduction is low, we show that USAT captures many properties possessed by $D^{P}$ many-one complete sets. We show that the structure of USAT can affect the structure of $D^{P}$ and the entire Polynomial Hierarchy (PH) as well. That is, 1. if USAT $\equiv^{P}_{m} \overline{USAT}$, then $D^{P}$ = co-$D^{P}$ and PH collapses. 2. if USAT $\in$ co-$D^{P}$, then PH collapses. 3. if USAT is closed under disjunctive reductions, then PH collapses. The third result implies that the probability bound in the Valiant-Vazirani reduction cannot be amplified by repeated trials unless the Polynomial Hierarchy collapses. These results show that even sets complete under "weak" randomized reductions can capture properties of many-one complete sets.