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)
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