  Over the next decade, improvements in cosmological parameter constraints will
be driven by surveys of large-scale structure. Its inherent non-linearity
suggests that significant information will be embedded in higher correlations
beyond the two-point function. Extracting this information is extremely
challenging: it requires accurate theoretical modelling and significant
computational resources to estimate the covariance matrix describing
correlations between different Fourier configurations. We investigate whether
it is possible to reduce the covariance matrix without significant loss of
information by using a proxy that aggregates the bispectrum over a subset of
Fourier configurations. Specifically, we study the constraints on $\Lambda$CDM
parameters from combining the power spectrum with (a) the modal bispectrum
decomposition, (b) the line correlation function and (c) the integrated
bispectrum. We forecast the error bars achievable on $\Lambda$CDM parameters
using these proxies in a future galaxy survey and compare them to those
obtained from measurements of the Fourier bispectrum, including simple
estimates of their degradation in the presence of shot noise. Our results
demonstrate that the modal bispectrum performs as well as the Fourier
bispectrum, even with considerably fewer modes than Fourier configurations. The
line correlation function has good performance but does not match the modal
bispectrum. The integrated bispectrum is comparatively insensitive to changes
in the background cosmology. We find that adding bispectrum data can improve
constraints on bias parameters and the normalization $\sigma_8$ by up to 5
compared to power spectrum measurements alone. For other parameters,
improvements of up to $\sim$ 20% are possible. Finally, we use a range of
theoretical models to explore how the sophistication required for realistic
predictions varies with each proxy. (abridged)
