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Safety in dynamical systems is commonly pursued using control barrier functions (CBFs) which enforce safety-constraints over the entire duration of a system's evolution. We propose a prescribed-time safety (PTSf) design which enforces safety only for a finite time of interest to the user. While traditional CBF designs would keep the system away from the barrier longer than necessary, our PTSf design lets the system reach the barrier by the prescribed time and obey the operator's intent thereafter. To emphasize the capability of our design for safety constraints with high relative degrees, we focus our exposition on a chain of integrators where the safety condition is defined for the state furthest from the control input. In contrast to existing CBF-based methods for high-relative degree constraints, our approach involves choosing explicitly specified gains (instead of class $\mathcal{K}$ functions), and, with the aid of backstepping, operates in the entirety of the original safe set with no additional restriction on the initial conditions. With Quadratic Programming (QP) being employed in the design, in addition to backstepping and CBFs with a PTSf property, we refer to our design as a QP-backstepping PT-CBF design. For illustration, we include a simulation for the double-integrator system.