How to identify causality, stability and ROC from the pole-zero plot?
For systems with more than one pole (with different radii), there are $3$ types of ROCs:


inside the circle with a radius corresponding to the smallest pole radius; the corresponding time-domain sequence is left-sided
outside the circle with a radius corresponding to the largest pole radius; the corresponding time-domain sequence is right-sided
in a ring defined by the radii $r_1$ and $r_2$ of any two poles with $r_1&lt;r_2$, with no other pole at a radius in the interval $(r_1,r_2)$; the corresponding time-domain sequence is two-sided


In your example you have case $3$ above, because that's the only ROC that includes the unit circle, and, consequently, corresponds to a stable system. Both other possible ROCs (inside the circle with radius $0.5$ and outside the circle with radius $2$) do not include the unit circle, and, consequently, do not correspond to impulse responses of stable systems. Since a ROC equal to a ring between two poles corresponds to a two-sided sequence, the system can't be causal. A causal system has a ROC outside a circle (case $2$ in the list above).