  Patchy reionization generates kinematic Sunyaev-Zeldovich (kSZ) anisotropies
in the cosmic microwave background (CMB). Large-scale velocity perturbations
along the line of sight modulate the small-scale kSZ power spectrum, leading to
a trispectrum (or four-point function) in the CMB that depends on the physics
of reionization. We investigate the challenges in detecting this trispectrum
and use tools developed for CMB lensing, such as realization-dependent bias
subtraction and cross-correlation based estimators, to counter uncertainties in
the instrumental noise and assumed CMB power spectrum. We also find that both
lensing and extragalactic foregrounds can impart larger trispectrum
contributions than the reionization kSZ signal. We present a range of
mitigation methods for both of these sources of contamination, validated on
microwave-sky simulations. We use ACT DR6 and Planck data to calculate an upper
limit on the reionization kSZ trispectrum from a measurement dominated by
foregrounds. The upper limit is about 50 times the signal predicted from recent
simulations.
