  We investigate how well the SPHEREx all-sky survey can constrain local
primordial non-Gaussianity beyond the parameter $f_{\text{NL}}$ using galaxy
power spectra. We forecast joint constraints on the parameters $f_{\text{NL}}$,
$g_{\text{NL}}$ and $\tau_{\text{NL}}$ obtained assuming a simple two-field
curvaton model of inflation. The parameters $f_{\text{NL}}$ and $g_{\text{NL}}$
characterise the squeezed limits of the primordial bispectrum and trispectrum
respectively, and lead to a characteristic scale-dependence of the galaxy bias
that increases out to arbitrarily large scales. Values of the parameter
$\tau_{\text{NL}}> (\frac{6}{5}f_{\text{NL}})^{2}$ cause the galaxy power
spectrum to have a stochastic component which also increases out to arbitrarily
large scales. Our MCMC forecasts indicate that SPHEREx can provide joint
constraints on any two of the three parameters $f_{\text{NL}}, \ g_{\text{NL}}$
and $\tau_{\text{NL}}$. Due to strong degeneracies among these parameters,
measurements of the galaxy power spectra alone may not be sufficient to jointly
constrain all three. Constraints on $f_{\text{NL}}, \ g_{\text{NL}}$ and
$\tau_{\text{NL}}$ obtained from galaxy power spectrum observations depend on
the modelling of underlying nuisance parameters. We study the robustness of our
forecast constraints to modelling choices and note that even with relatively
conservative modelling assumptions, SPHEREx galaxy power spectra can provide
strong evidence of local non-Gaussianity, even if the particular values of
$f_{\text{NL}}$ and $g_{\text{NL}}$ cannot be measured precisely.
