  We propose three physical tests to measure correlations in random numbers
used in Monte Carlo simulations. The first test uses autocorrelation times of
certain physical quantities when the Ising model is simulated with the Wolff
algorithm. The second test is based on random walks, and the third on blocks of
n successive numbers. We apply the tests to show that recent errors in high
precision simulations using generalized feedback shift register algorithms are
due to short range correlations in random number sequences. We also determine
the length of these correlations.
