  Dirac and Weyl semimetals both exhibit arc-like surface states. However,
whereas the surface Fermi arcs in Weyl semimetals are topological consequences
of the Weyl points themselves, the surface Fermi arcs in Dirac semimetals are
not directly related to the bulk Dirac points, raising the question of whether
there exists a topological bulk-boundary correspondence for Dirac semimetals.
In this work, we discover that strong and fragile topological Dirac semimetals
exhibit 1D higher-order hinge Fermi arcs (HOFAs) as universal, direct
consequences of their bulk 3D Dirac points. To predict HOFAs coexisting with
topological surface states in solid-state Dirac semimetals, we introduce and
layer a spinful model of an $s-d$-hybridized quadrupole insulator (QI). We
develop a rigorous nested Jackiw-Rebbi formulation of QIs and HOFA states.
Employing $ab\ initio$ calculations, we demonstrate HOFAs in both the room-
($\alpha$) and intermediate-temperature ($\alpha''$) phases of Cd$_{3}$As$_2$,
KMgBi, and rutile-structure ($\beta'$-) PtO$_2$.
