Okay, I thought about it again, and I realized that neither the money, nor the person, needs to go back in time. The trick is that the only thing that needs to go back in time, is the information. No 'time travel' necessary. The 'time machine' in the QuickerThanQuickCash store branches is just a computer linked to computers in the past. And in fact, since the computer data connections and technology would all be made originating in the present and extending into the future, for future use, no changes would ever have to be made 'in the past'. It would all be 'from this day forward'.
That is, the future 'you' deposits money through automatic banking into your account today. It is all done by computer. Only the information about the deposit has to be made 'retroactively'.
If there are enough QuickerThanQuickCash store outlets, then everyone would know about it. They would not be suprised that money showed up in their account, as everyone would be doing it. In fact, you might expect to get cash from your future 'you', just like today you might get cash form a relative, depositied into your account, without expecting it. The money would just show up. But just because 'you' got the cash today, 'you' would still have to go into the QuickerThanQuickCash store tomorrow to arrange the loan. Failure to do so means that the cash does not show up in your account today.
It's all about information and data traveling through time, not physical objects.
Even though 'you' got the money today, the loan is still made tomorrow, and the documents are signed tomorrow. 'You' are still legally accountable for the documents you sign tomorrow, in all of the days after tomorrow. There is always only one 'you' at any one time, and there is always only one timeline. The only real difference between on-line banking under the reality of today and this scenario, is that 'you' would just assume the money that showed up in your account today was a result of 'you' taking out a loan tomorrow, instead of some mysterious benefactor puting the money into your accounnt today. In fact, the deposit could have a notation that indicates it was a deposit from the future. Interest would be paid from the date of deposit, of course, by contract, but 'you' would be well aware of this when 'you' took out the loan, 'you' agreed to it at the time the contract was made. It would be entirely legal and enforceable from tomorrow on through eternity. It would not, of course, be enforceable 'today', because the contract was not signed 'today', but this is of no consequence to QuickerThanQuickCash store in the future. They really do not care about today, that is in the past.
Really, it would be like sending a money transfer to another person today, except that it would arrive yesterday, and the money would be available yesterday, not today. Since it is not really physical money, but a number on a spreadsheet, there would not be any timeline paradox. When 'you' take out the loan tomorrow, 'you' already know it has already shown up in 'your' account today, 'you' just did not know today that 'you' took out the loan tomorrow, until it shows up. Nothing really changes in time. Nothing changes in the past, because of a change made today, so nothing changes today because of a change made by someone in the future going back into the past. Just an accounting procedure made retroactively.
It does play havoc with our current notions of 'causality', and about things in the future causing things to happen today, but this is sci fi, after all. Granger causality is in some ways just as weird. But still, even though it happened today because of something in the future, it still happened only in our timeline, and our timeline only occured once. What happens, happens, irregardless of when it is caused.
I am not really sure what the economy would look like, when money could routinely be depositied into an account in the past, but it would be interesting. I suspect, at the least, that convention would demand no money would ever be transferrred back into the past beyond the time that the technology was first made available, because that would really botch up the bookkeeping. In fact, since the technology did not exist further back in time than when it was developed, I see no way money could be transferred back before the technology was implemented. But it would be fun to hope, today, that someone from the future just might send you money today.