  Consider a scenario where a quantum particle is initially prepared in some
bounded region of space and left to propagate freely. After some time, we
verify if the particle has reached some distant target region. We find that
there exist "ultrafast" ("ultraslow") quantum states, whose probability of
arrival is greater (smaller) than that of any classical particle prepared in
the same region with the same momentum distribution. For both projectiles and
rockets, we prove that the quantum advantage, quantified by the difference
between the quantum and optimal classical arrival probabilities, is limited by
the Bracken-Melloy constant $c_{bm}$, originally introduced to study the
phenomenon of quantum backflow. In this regard, we substantiate the
$29$-year-old conjecture that $c_{bm}\approx 0.038$ by proving the bounds
$0.0315\leq c_{bm}\leq 0.072$. Finally, we show that, in a modified projectile
scenario where the initial position distribution of the particle is also fixed,
the quantum advantage can reach $0.1262$.
