  Plasma accelerators driven by intense laser or particle beams provide
gigavolt-per-meter accelerating fields, promising to drastically shrink
particle accelerators for high-energy physics and photon science. Applications
such as linear colliders and free-electron lasers (FELs) require high energy
and energy efficiency, but also high stability and beam quality. The latter
includes low energy spread, which can be achieved by precise beam loading of
the plasma wakefield using longitudinally shaped bunches, resulting in
efficient and uniform acceleration. However, the plasma wavelength, which sets
the scale for the region of very large accelerating fields to be 100 {\mu}m or
smaller, requires bunches to be synchronized and shaped with extreme temporal
precision, typically on the femtosecond scale. Here, a self-correction
mechanism is introduced, greatly reducing the susceptibility to jitter. Using
multiple accelerating stages, each with a small bunch compression between them,
almost any initial bunch, regardless of current profile or injection phase,
will self-correct into the current profile that flattens the wakefield, damping
the relative energy spread and any energy offsets. As a consequence, staging
can be used not only to reach high energies, but also to produce the exquisite
beam quality and stability required for a variety of applications.
