  Clustering of large-scale structure provides significant cosmological
information through the power spectrum of density perturbations. Additional
information can be gained from higher-order statistics like the bispectrum,
especially to break the degeneracy between the linear halo bias $b_1$ and the
amplitude of fluctuations $\sigma_8$. We propose new simple, computationally
inexpensive bispectrum statistics that are near optimal for the specific
applications like bias determination. Corresponding to the Legendre
decomposition of nonlinear halo bias and gravitational coupling at second
order, these statistics are given by the cross-spectra of the density with
three quadratic fields: the squared density, a tidal term, and a shift term.
For halos and galaxies the first two have associated nonlinear bias terms $b_2$
and $b_{s^2}$, respectively, while the shift term has none in the absence of
velocity bias (valid in the $k \rightarrow 0$ limit). Thus the linear bias
$b_1$ is best determined by the shift cross-spectrum, while the squared density
and tidal cross-spectra mostly tighten constraints on $b_2$ and $b_{s^2}$ once
$b_1$ is known. Since the form of the cross-spectra is derived from optimal
maximum-likelihood estimation, they contain the full bispectrum information on
bias parameters. Perturbative analytical predictions for their expectation
values and covariances agree with simulations on large scales, $k\lesssim
0.09h/\mathrm{Mpc}$ at $z=0.55$ with Gaussian $R=20h^{-1}\mathrm{Mpc}$
smoothing, for matter-matter-matter, and matter-matter-halo combinations. For
halo-halo-halo cross-spectra the model also needs to include corrections to the
Poisson stochasticity.
