  We study the general properties of the CMB temperature four-point function,
specifically its harmonic analogue the angular trispectrum, and illustrate its
utility in finding optimal quadratic statistics through the weak gravitational
lensing effect. We determine the general form of the trispectrum, under the
assumptions of rotational, permutation, and parity invariance, its estimators
on the sky, and their Gaussian noise properties. The signal-to-noise in the
trispectrum can be highly configuration dependent and any quadratic statistic
used to compress the information to a manageable two-point level must be
carefully chosen. Through a systematic study, we determine that for the case of
lensing a specific statistic, the divergence of a filtered temperature-weighted
temperature-gradient map, contains the maximal signal-to-noise and reduces the
variance of estimates of the large-scale convergence power spectrum by over an
order of magnitude over previous gradient-gradient techniques. The total
signal-to-noise for lensing with the Planck satellite is of order 60 for a LCDM
cosmology.
