In macroeconomics, there is a term called “neutrality of money” which basically says you can’t affect GDP with open market operations (in the game this most closely translates to the QE and Helicopter Money policies). The reason is because of what economists call the real-nominal principle. While blowing up inflation will nominally grow GDP (the literal size will increase), real GDP is balanced by inflation.
I will note that there are some studies in econometrics that suggest monetary policy can have long-term effects on GDP variables (consumption, notably). The reason is explained in the Wikipedia article as relating to “super neutrality” which basically translates to the growth rate of the money supply (basically just the inflation rate) causing investment expenditures to go down and durable consumer expenditure to go up (durables = consumption spending on things that last, like refrigerators). I recommend trying to find some econometrics studies that attempt to model this effect if you want to have inflation affect GDP in the game. Otherwise, it still just blatantly economically incorrect.
In the case of QE, it’s simply altering long term interest rates to increase the supply of money in the short run. Again, due to the neutrality of money, this policy very likely has no effect on GDP because as I explained, inflation balances it out exactly (in the long run, when real GDP is something you can even talk about. In the short run, you do get a nominal GDP blip, but that is a worthless statistic).
Ah, I didn’t know that helicopter money is actually a specific term existing outside the context of the game. After doing some research, it appears to be the case that the policy is mathematically equivalent to a combination of expansionary fiscal policy and expansionary monetary policy. The former crowds out the economy so real GDP should actually shrink since investment expenditures go down (again, in the long run). However the expansionary monetary aspect will have no real effect due to the neutrality of money; again, inflation and nominal GDP balance each other out here.
So what you’re saying is that GDP should quickly rise right away, but then with a delay, fall by just as much as it rose before, making it useless for long term but potentially useful if you really just need to quickly boost your stats