Answer:
Left turns are either permissive (no arrow), exclusive/permissive (allowing turns on gaps in traffic after the arrow expires, usually with a five-section display) or exclusive- only (no turns after the arrow expires, three-section display).
Additional: At those intersection controlled by green turn arrows - there is no uniformity among the states as to whether the left turn arrow leads the regular light sequence or follows it. That decision is usually left to the Highway Dept. traffic engineers.
More: The choice of leading left turn or lagging (following) left turn is not usually a policy of the highway department, but is an engineering decision made separately for each intersection. Often, to achieve signal progression (lights turn green as you come to them), the choice of leading or lagging left turn is forced by the progression plan. Often, to make the progression work better on a two-way street at a particular intersection, one direction on the street has a leading left turn, but the opposite direction has a lagging left turn. This way, the cars from each direction can arrive at this intersection at different times.
The lagging left turn has a disadvantage. If left turns are permitted on a circular green, a lagging left turn in one direction causes a hazard: One direction on the street receives circular yellow, but the opposite direction remains green. A driver receiving the yellow might turn left across live traffic moving on a green, thinking they will stop. To prevent this, either the signal with a lagging left turn must prohibit left turns on circular green, or use the new flashing yellow arrows.
A flashing yellow arrow display uses a flashing yellow arrow instead of a circular green, and has a 4-section display with all arrows.