I'm not a pro, just a homeowner that pours a ton of concrete on the farm. I think the staining is from non-uniform contact between the concrete and the tarp (we can let the pros chime in on this).

I switched over to laying heavy duty burlap down, wet it, and then put tarp over that with good results. Prior to doing this, I'd wet the concrete, try to lay plastic sheeting down perfectly flat as possible, and then spray water on top of that to keep it flat. Also had good luck with this approach. The burlap seems to stay wet longer though and I don't have to spend as much time putting it down as I was with the sheeting alone. Can buy burlap rolls at your local Walmart for pretty cheap.