What kind of power spectrum implies uncorrelation in time?
Since this does seem to answer the question, I'm reposting my comment as an answer:

I guess you may have to distinguish continuous (analog) and discrete (digital) signals. For analog signals, the autocorrelation function is a Dirac if and only if the spectrum is flat for all frequencies. Typically though what we care about is whether or not our digital samples are uncorrelated. It is easier to safisfy this. If your low-pass filter before your ADC is, say an ideal one, your ACF will be a sinc. If you get things right, your samples are in the zero crossings of the sinc and the discrete ACF looks like a discrete delta...

As the OP mentioned, the trick is to sample the signals in the zero crossings of the sinc function. For an ideal (rectangular) lowpass filter with (two-sided) bandwidth $B$ this means sampling exactly at a rate $1/B$.