In my study, I have 4 quantiles and to ascertain the statistically significant difference in systolic blood pressure across the quantiles, I did a Kruskal Wallis test in Stata (Medians; 1116, 119, 120 and 127, p<0.001). What is the best multiple comparison method for me to use in order to determine which groups were statistically significant?

Thanks for a great job from this group, I have really benefited a lot as it relates to the appropriate application of statistical methods.

Information-losing transformations of variables is seldom a good idea. Quantile groups are piecewise flat transformations of inherently continuous variables. I recommend modeling Y=SBP using raw data for X.

The model that contains the K-W test as a special case is the proportional odds ordinal logistic semiparametric model. Whatever you decide to do with X, the PO model will handle Y without making a distributional assumption for Y given X=x.

Thank you for the advice, but is there a posthoc test for Kruskal Wallis test?

You must use the proportional odds model to get rational post hoc tests. The model likelihood ratio \chi^2 test from the PO model provides the overall test for k groups, and then do any contrasts of interest using the estimated regression coefficients and their standard errors and covariances. This is laid out in https://hbiostat.org/doc/bbr.pdf in the Nonparametrics chapter. The *score* \chi^2 test from the PO model is the K-W test.