Let's say government can spend $100 billion today or spend the present expected value of $100 billion, stretched out over time so it is a commitment in perpetuity. Both spending programs are financed by bonds. So that's the same net present value of spending and the same method of finance.
The Keynesian boost to aggregate demand arises because people consider the resulting bonds to be "net wealth" even when they are not, in the sense outlined by Robert Barro (1974). People are tricked by the government's fiscal policy, but of course the extent, timing, and nature of the trickery is hard to predict.
Is it easier to trick people "a lot all at once" or "a little bit by bit over time"? It depends. If you try to trick them slowly over time, temporal learning and adaptive expectations may work against the policymaker. But if you try to trick people a lot all at once, the trick may rise over their threshold of attention, perhaps because of media coverage. We don't know which "trick" to aggregate demand will be greater, the temporary boost to spending or the permanent boost.
One way to get a clear answer — in favor of Krugman's hypothesis that the temporary spending is more potent — is to assume that bonds as net wealth fails for a policy rule but not for a single period policy surprise (the temporary boost in spending). In contrast, the traditional Keynesian view is to think that bonds are also net wealth in the medium run and perhaps the long run too and then we are back to not knowing whether the permanent or temporary spending boost does more for aggregate demand. Or you might think, as I have suggested, that whether bonds are viewed as net wealth in the short run will depend on the size of the spending boost. Many different assumptions are possible and thus many different results are possible.
Alternatively, you might compare $100 billion today (and no more) to $100 billion each year, every year. You could call that "temporary" vs. "permanent" although I suspect the dominant effects will fall out of "small" vs. "large."
The latter, permanent boost to spending will give a bigger boost to aggregate demand overall (unless again you neuter it by applying Ricardian Equivalence to the rule but not the single period policy). It also will lead to more crowding out. Do note that in the early periods of this policy taxes need not rise by $100 billion for each year but rather the early installments can be paid off over time.
It is less clear whether the permanent spending boost leads to a bigger AD shift only for today. It will if you apply the same degree of bonds as net wealth to the rule and single period policy, and if you think that the later periods of government spending will add net value, thus creating positive feedback through the long-run wealth effect.
It is also unclear if the larger, permanent spending boost creates more "stimulus per dollar" (as opposed to more stimulus in the aggregate or more stimulus for the single period). That will depend on whether we are in the range where the stimulus has increasing returns to scale (maybe a certain critical mass is needed, as I believe Mark Thoma has suggested), constant returns to scale, or diminishing or even negative returns to scale, because of eventual crowding out.
Overall the Keynesian effects can mean either the permanent or the temporary spending boost has a bigger effect and there are also a number of ways of defining what a "bigger effect" might mean. This analysis has more variations than does the Poisoned Pawn Sicilian.