  We study reconstruction of the lensing potential power spectrum from CMB
temperature data, with an eye to the Planck experiment. We work with the
optimal quadratic estimator of Okamoto and Hu, which we characterize thoroughly
in application to reconstruction of the lensing power spectrum. We find that at
multipoles L<250 our current understanding of this estimator is biased at the
15% level by beyond-gradient terms in the Taylor expansion of lensing effects.
We present the full lensed trispectrum to fourth order in the lensing potential
to explain this effect. We show that the low-L bias, as well as a previously
known bias at high-L, is relevant to the determination of cosmology and must be
corrected for in order to avoid significant parameter errors. We also
investigate the covariance of the reconstructed power, finding broad
correlations of ~0.1%. Finally, we discuss several small improvements which may
be made to the optimal estimator to mitigate these problems.
