  Gravitational-wave tails are linear waves that backscatter on the curvature
of space-time generated by the total mass-energy of the source. The non-linear
memory effect arises from gravitational waves sourced by the stress-energy
distribution of linear waves themselves. These two effects are due to quadratic
multipolar interactions (mass-quadrupole and quadrupole-quadrupole) and are
well known. Also known are the tails generated by tails themselves (cubic
"tails-of-tails") and the tails generated by tails-of-tails or vice versa
(quartic "tails-of-tails-of-tails"). In this work, we focus on the cubic
"tails-of-memory" corresponding to the mass-quadrupole-quadrupole interaction,
as well as the "spin-quadrupole tails", which are due to the cubic interaction
between the mass, the total angular momentum and the quadrupole. The
tails-of-memory and the spin-quadrupole tails contribute to the asymptotic
waveform at the fourth-post-Newtonian (4PN) order beyond quadrupolar radiation.
