In summary, using the state of the art PIGLET simulations, we have studied temperature dependent double proton transfer in molecular crystals of terephthalic acid. In accordance with experiments, we observe occurrence of proton transfers and thereby onset of orderdisorder transitions at temperatures as low as 70 K. At around 300 K the H atoms are completely disordered. Our simulations show that, though the neutral L and R tautomers are the dominant species, there is a small but non-negligible amount of charged species present at both low and high temperatures. The simulations suggest that while between 70 and 200 K, the proton transfer happens through tunnelling, at room temperature it most probably occurs through activated hopping. However, we note that in order to conclusively comment on the mechanism through which the transfer events occur, one needs to perform further simulations using enhanced simulation techniques because these are rare events requiring also the identification of the relevant reaction coordinates. Additionally, in accordance with other systems, we observe that NQE reduces the coupling between the proton transfer and reduction of the donor-acceptor heavy atom distances. Further, through the calculations of the Wannier functions, we show that the H atom is transferred in form of a proton from the donor to the acceptor O atom.