Normal score transformation This question is related to a normal score transformation which is performed in the paper by
RUAN, Quansong, et al. Local similarity analysis reveals unique associations among marine bacterioplankton species and environmental factors. Bioinformatics, 2006, 22.20: 2532-2538.

They claim that given a variable X with observations $x= (x_1,\ldots,x_n)$ that are not normally distributed, normal score transformation is performed by first creating a rank vector observations $x$, $R^x = (R_1^x,\ldots, R_n^x)$ and then using and then transforming to $x'=(x_1',\ldots,x_n)$, by $x_i\ = \phi_{-1} (\frac{R_i^x}{n+1} )$ where $\phi_{-1}$ is the inverse of cummulative normal distribution. 

It's intuitively clear (but maybe a bit harder to prove?) that resulting variable $X'$ has observations normally distributed, but I have a question about time series dependence. If we had that observations $x$ were correlated, say, over 4 adjacent observations, , i.e. $x_{n+1} = f(x_{n},x_{n-1},x_{n-2})$, how does this change after the normal transformation? Can we still find a function g, so that $x'_{n+1} = g(x'_{n},x'_{n-1},x'_{n-2})$ would hold? Since we're transforming based on the rank of data, I expect this to be not possible?
As you suspected, correlations will be heavily changed by such a transformation. The reason for this is also exactly as you described, namely that using ranks rather than values for the transformation eliminates all of the information needed to actually preserve these relationships. As an example, suppose we have a single observation with rank 1 across all observations that none the less shows a decrease in absolute value across them (i.e., there's a correlation with a negative slope over time). Such a transformation will result in a correlation of 0.

One might be able to find a different transformation that preserves this relationship, or might be able to simply pool all of the time points together before computing the ranks (this would then at least partially preserve some correlations, though depending on what's needed to properly normalize the data it may cause other problems), but in general there's no guarantee that this is possible.