In this letter we present constraints on the scale-dependent ‘ ‘ local ’ ’ type primordial non-Gaussianity , which is described by non-Gaussianity ’ s spectral index n _ { NG } , from the NRAO VLA Sky Survey and the quasar catalog of the Sloan Digital Sky Survey ( SDSS ) Data Release 6 , together with the SDSS Data Release 12 photo-z sample . Here , we use the auto-correlation analyses of these three probes and their cross-correlation analyses with the cosmic microwave background ( CMB ) temperature map , and obtain the tight constraint on the spectral index : n _ { NG } = 0.2 ^ { +0.7 } _ { -1.0 } ( 1 \sigma C.L . ) , which shows the first competitive constraint on the running of non-Gaussianity from current large-scale structure clustering data . Furthermore , we also perform the forecast calculations and improve the limit of n _ { NG } using the future Euclid mission , and obtain the standard deviation at 68 % confidence level : \Delta n _ { NG } = 1.74 when considering the fiducial value f _ { NL } = 3 , which provides the complementary constraining power to those from the CMB bispectrum information .