In this paper, we consider the unconstrained submodular maximization problem. We propose the first algorithm for this problem that achieves a tight $(1/2-\varepsilon)$-approximation guarantee using $\tilde{O}(\varepsilon^{-1})$ adaptive rounds and a linear number of function evaluations. No previously known algorithm for this problem achieves an approximation ratio better than $1/3$ using less than $Ω(n)$ rounds of adaptivity, where $n$ is the size of the ground set. Moreover, our algorithm easily extends to the maximization of a non-negative continuous DR-submodular function subject to a box constraint and achieves a tight $(1/2-\varepsilon)$-approximation guarantee for this problem while keeping the same adaptive and query complexities.