  We study the covariance in the angular power spectrum estimates of CMB
fluctuations when the primordial fluctuations are non-Gaussian. The
non-Gaussian covariance comes from a nonzero connected four-point correlation
function -- or the trispectrum in Fourier space -- and can be large when
long-wavelength (super-CMB) modes are strongly coupled to short-wavelength
modes. The effect of such non-Gaussian covariance can be modeled through
additional freedom in the theoretical CMB angular power spectrum and can lead
to different inferred values of the standard cosmological parameters relative
to those in $\Lambda$CDM. Taking the collapsed limit of the primordial
trispectrum in the quasi-single field inflation model as an example, we study
how the six standard $\Lambda$CDM parameters shift when two additional
parameters describing the trispectrum are allowed. The reduced statistical
significance of the Hubble tension in the extended model allows us to combine
the {\it Planck} temperature data and the type Ia supernovae data from
Panstarrs with the distance-ladder measurement of the Hubble constant. This
combination of data shows strong evidence for a primordial trispectrum-induced
non-Gaussian covariance, with a likelihood improvement of $\Delta \chi^2
\approx -15$ (with two additional parameters) relative to $\Lambda$CDM.
