HUP for spins reads
$$\langle\sigma_z^2\rangle\langle\sigma_x^2\rangle \ge \frac{1}{4}|\langle\sigma_y\rangle|^2$$
Right after measuring ##\sigma_z##, we know it exactly, and so ##\langle\sigma_z^2\rangle=0##.
However, HUP then implies that ##\langle\sigma_y^2\rangle=\infty##
Even if we say...