i have only been watching nfl for a couple of years but as my understanding goes there are no limitations on a qb crossing the line of scrimmage with the exception that should he cross the line of scrimmage before the ball is snapped his team will inccur a penalty. but as i say he may cross the line for a draw play for instance or on a trick play or just if he sees no alternative though once he is past the line of scrimmage (his entire body pads an all) he may not make a forward pass only backwards or lateral like any other player on the field
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whenever he has the ball and is in on offense, but he would probably run if he cant get anyone open to pass to and sees an open gap he can run through to get some yards.