  We present quantum Monte Carlo results for the field and temperature
dependence of the magnetization and the spin-lattice relaxation rate $1/T_1$ of
a two-dimensional $S=1/2$ quantum Heisenberg ferromagnet. The Monte Carlo
method, which yields results free of systematic errors, is described in detail.
The high accuracy of the calculated magnetization allows for stringent tests of
recent approximate analytical calculations. We also compare our results with
recent experimental data for a $\nu=1$ quantum Hall ferromagnet, which is
expected to be well described by the Heisenberg model. The dynamic response
function needed to extract $1/T_1$ is obtained using maximum-entropy analytic
continuation of the corresponding imaginary-time dependent correlation
function. We discuss the reliability of this approach.
