A processor of an aspect includes a decode unit to decode a user-level suspend thread instruction that is to indicate a first alternate state. The processor also includes an execution unit coupled with the decode unit. The execution unit is to perform the instruction at a user privilege level. The execution unit in response to the instruction, is to: (a) suspend execution of a user-level thread, from which the instruction is to have been received; (b) transition a logical processor, on which the user-level thread was to have been running, to the indicated first alternate state; and (c) resume the execution of the user-level thread, when the logical processor is in the indicated first alternate state, with a latency that is to be less than half a latency that execution of a thread can be resumed when the logical processor is in a halt processor power state.