Why do we remember the past but not the future? The question is sometimes referred to as the "psychological arrow of time" (Hawking, 1985). Here the past is understood as a moment or time when the entropy of the universe was lower, and contrarily for the future. So it is generally thought that PAOT is a consequence of the thermodynamic arrow of time of our universe. If so (maybe not?), how do the two relate? 
Some explanations in the literature:


*

*Practical memory systems work in a way that the formation of new memories entails an overall increase of total entropy of the system and the environment. For example, to create a memory, i.e. to cause our neurons to orient in a particular fashion, requires energy which results in our body heating up a little bit, increasing the total entropy (Hawking, 1985 and 1994); The initialization of memory to make it reusable is an irreversible process that increases total entropy (Landauer, 1961. Wolpert, 1992).

*More recently, People have argued that even reversible and non-dissipative memory systems are subject to PAOT (Mlodinow and Brun, 2014). The conclusion is arrived by imposing some constraints on what a memory system should be like. Specifically, they argue that a memory should be somehow robust to small microscopic changes in states of the system it records (what they call "generality" requirement). But the smallest changes in the future state destroy the thermodynamic arrow of time between now and the future. So any memory of the future of the system "could remember only one possible configuration of that system". This fine-tuning disqualifies it as a bona fide memory.


My problem with explanation (1) is that even if it's correct, it doesn't seem to be a complete answer in itself. Yes, increase of (new) memory happens only as total entropy of the universe increases. So what? It doesn't have anything to say on the nature of that memory. Why couldn't it occasionally be a memory of the future for that matter? Explanation (2) leaves no such ambiguity. But the generality requirement seems artificial: surely a memory that records the only future configuration of the system remembers the future in a deterministic world, there being no "what ifs" regarding that state? 
Of course, my understanding of the problem is only preliminary. I'd like to know whether there is not some generally accepted explanation, or any other thoughts you have on it. 
 A: It is a reasonable  question at the  elementary particle physics level , since the mathematical formulae of all the models we have are  reversible as to time. It is in the thermodynamic manifestation of the laws that an arrow of time appears, and in special relativity which separates observations in timelike and spacelike regions.
So it is one of those questions of "why" addressed to physics that really have the only answer "because". Macroscopically , where we live and die, there exists the arrow of time described by the laws of thermodynamics, and that is that.
The theory of special relativity has been validated with innumerable measurements and therefor again the answer is "because this is what we observe. A future event cannot register in the present because of the velocity of light which is the limit in the transfer of information.
Edit for clarification:
There are theories for  particle physics extending the standard model, and there tachyons can exist. Tachyons in these theories can travel backwards in time and could interact with the electromagnetic structure of our brain if the theory allows it. Thus it could affect the  memory banks and leave a record that could be interpreted as a future memory. There is no experimental sign for such particles.
So future memories belong to science fiction and metaphysics (precognition and such).
A: If you think of the future as a probabilistic distribution of events, for the far future there are an infinite number of possible events. As you approach those events in time, past (and present) actions force the future to collapse to a single event (assuming two can't happen simultaneously). 
You could think (and even predict) one event would happen over another, but you can't have a memory of one because it hasn't happened until it happens. 
In this respect, you could almost think of events in time as being a larger version of Schrödinger's cat
A: All you have direct access to at any moment is the macrostate of your brain at that moment.   A (backward) memory is an inference from that state to what the macrostate of the world was at some time in the past and a (forward) prediction is an inference from that state to what the macrostate of the world will be at some time in the future.
A given macrostate is compatible with a great many microstates, each of which yields inferences about the past and inferences about the future, all by running the same laws of physics both backward and forward.  This gives you a range of possibilities (and a probability distribution) for what the past was like and a range of possibilities (and a probability distribution) for what the future will be like.
So I  think your question should be rephrased this way:  Why are we so much more confident about our backward inferences than about our forward inferences?  The answer must be twofold:  


*
*
The inferred probability distribution over past states is much more concentrated than the probability distribution over future states.  (That is, our memories of the past are more accurate than our memories of the future.)

*At some level, we are aware of this. (That is, we give far greater credence to our memories of the past than our memories of the future.)

The first follows from increasing entropy, which in turn follows from the fact that the Universe was once in a state of very low entropy.  The second follows, perhaps, from the first together with natural selection, which rewards true beliefs and punishes false ones.
So I think the answer to your question is something like this:  Our ability to remember the past but not the future (or more precisely, our much greater faith in our backward predictions than in our forward predictions) results from Darwinian evolution in a universe that started with a Big Bang.
A: The answer to your question is that we don't remember the future because we haven't yet stored any memory of it. 
Your memories arise from connections wired in your brain as a consequence of experience. You have not yet experienced the future, so the configuration of your brain has not been affected by it. 
You can, of course anticipate and imagine the future. Indeed, the sensation of imagining, say, a walk in the woods tomorrow, can be as vivid as recalling a walk in the woods a year ago.
The phenomenon is not in the least confined to the human mind. The history file associated with my browser contains only the sites I have already visited. It does not contain information about the sites I am going to visit, because I haven't yet done so.
The one way direction of time arises naturally if you consider time to be a counter of change (as it is in the case of SI units). If I increment a counter every time there is a transition between the states of a given caesium atom the counter will inexorably rise. If the transitions were to somehow reverse themselves they would still count as changes, and the total on the counter would still increase. The count can never decrease.
To take another example, the number of transactions on your bank account can never decrease, because, unlike the balance of the account, it does not differentiate between positive payments in and negative payments out- it is counts both as an increase regardless of their sign. Past transactions are those that have been counted- future transaction have yet to be counted.
There should be no mystery to the arrow of time. Change happens. Time is simply a count of it. 
A: This seems like an extremely deep question. The brain is  a physical system. So it must have some mathematical object that describes its current state $S$. And this state evolves in time, so we have $S(t)$ representing the entire life of the brain.
Suppose we're given the state at some time $t_0$, $S(t_0) $. The question is, why are we able to extract more information from $S(t_0) $ about the past $S(t<t_0) $ than the future $S(t>t_0) $?. After all, the laws of the universe are time-symmetric. The future retro-causes the past as much as the past causes the future.
The only real asymmetry that I see is in wavefunction collapse. In the information theory interpretation, the wavefunction represents the knowledge of an agent about the universe. The collapse is knowledge becoming known.
In this sense, the future states $S(t>t_0) $ are fundamentally not extract-able from the present state $S(t_0) $
But then, the past states are not extractable from $S(t_0)$ for the exact same reason. It's well known that you can't know the past wavefunction from the collapsed wavefunction.
I believe we have to model $S$ as a cumulative quantity. $S(t_0) $ represents the cumulative knowledge gained from all the past knowledge gains. This way, the definition of $S$ treats time asymmetrically.
In short, we need two "axioms" :

*

*The laws of the universe are non-deterministic. This makes $S(t>t_0) $ non-extractable from $S(t_0) $


*$S(t) $ is a cumulativive quantity, accumulating in the direction past to future. This makes the past states extractable from $S(t_0) $
Cumulative quantities seem like they violate time-symmetry, but they don't. Think of a mountain that grows bigger and bigger with time. The laws behind this are obviously time-symmetric, but that doesn't mean that the phenomena have to be time-symmetric.
A: The reason we don't remember the future has to do with an arrow that gets ignored. That of information flow between the brain and the outer world. One could call it the sensory arrow of time. With our senses, we absorb information only in the future direction i.e. along the thermodynamic arrow of time. We never absorb information along the opposite direction of time. In other words we never emit it along the normal direction. We actually do emit information (speaking, touching) but not in the "future-memory' way. I will explain how emitting information is the key to the definition of future memories.
Imagine there were beings who somehow experienced time in reverse. They call the future past and the past future. Now from their point of view everything happens in reverse. Dissipated heat collects into light and goes back into stars, smell molecules waft back into roses, etc. However one should note that they can't observe these reverse-dissipation events.
For them it is us who are the future-remembering beings. How they "see" us can give us a clue to what future memory can look like to us. Now from our point of view, information from objects enters our bodies, gets stored as memories and finally decays / gets corrupted. But as per them, who live reverse lives, our memories arise randomly, reverse themselves and then the information from those memories (lights, fragrances, sounds) is sent out to the external world where it gets absorbed by objects.
Thus it follows that when we, the past remembering beings, want to talk of future memories, then it must be about forming memories, reversing and converting them into information and then sending that information out.
So, if we wanted to have a memory of the future, we could, with some physiological modifications, do something like this. We could imagine an object like a blue square and remember this imagination. Now we reverse this memory and emit blue photons from our body in such a way that this light forms a blue square on a white screen. This is a valid memory because it has a causal connection with the square and is also a memory of the future because the square came after the memory. Imagined in reverse the photons collect and fall on the white sheet, get reflected back in a blue stream, enter our body, get converted to nerve signals and are finally imprinted on our brains as a memory. This series of these events is not improbable because entropy decreases in the past here.
So just like past memories are 'decodable' impressions of past information, future memories must be decodable sources of future information.
This definition might seem problematic at first because this way you seemingly can't remember "complex" events like looking at a frog. But then even past memories are also always inferences. This has to do with the 2nd law of thermodynamics. It is very likely that information comes from objects than just randomly flowing into us in the shape of objects. So it is improbable that we will see frogs just because light rays collected in such a way by chance and impinged on our eyes.
Future-remembering beings would have a hard time keeping track of the world around them because generally information spreads out from objects. However if perhaps there were some chance events where local entropy was reversed like yellow painting which absorbed yellow and heat gave off white light, then their memories could make sense.
We also need to remember that in the end memories point to information transfer from the body to the brain and not from the outside world to the body. Touching something cold for example makes information from the body flow outward but it still sends impulses to the brain.
Remembering the future is not about time travel or knowing a future event without having any connection to it. And if there was indeed a way of knowing the future, it would still not mean having a memory of the future unless causality was involved.
This definition also avoids any paradoxes. You can't use this memory to change the future. If you do then the memory was simply false. Also, claiming to change the future by destroying a future memory would be akin to claiming to change the past by implanting a false past memory.
We don't remember the future because we don't / can't convert our memories into outside information in a way that is consistent with how we decode that memory in our heads. It makes zero evolutionary sense to align the sensory arrow against the thermodynamic arrow of time.
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My rather simple answer, which almost no one has paid heed to or understood, is that we can remember the future; it's just not worth it nor is it really spectacular. And there is nothing mystical about it. Hear me out; no pseudoscience here but quite a simple answer to an often asked question. Please ask me if you don't understand something about it.
So here it goes. You imagined a blue circle in your head and stored it as a memory. You then reversed (forgot) that imagination and that forgetting was the exact reversal of the process of remembering so that it led to emission of blue light from your body. That light produced an image of a blue circle on a white screen for an instant. Thus, that imagination was a memory of the future because you remembered it before it happened. If you reverse this whole process you can see how the blue photons from the fleeting circle flew into you and became your memory.
On the other hand if you imagined a blue circle and simply saw one the next day it would be chance and not a future memory because the blue circle was not formed by your memory.
Remembering the future doesn't mean knowing something you have no connection to like the winning number of the next lottery. lt is simply the reversal of the process of remembering (memorization reversal and sensory perception reversal).
Adding some points for further clarity -
Any valid memory must always have a causal connection with the corresponding information outside the memory housing body. And a mechanism should exist in that body to decode that memory because on their own memories don't represent anything. The definition of future memory here meets both these criteria.
In summary, just like past memories are 'decodable' impressions of past information, future memories must be decodable sources of future information.
So, we don't remember the future because we don't / can't convert our memories into outside information in a way that is consistent with how we decode that memory in our heads.
If you are downvoting please leave comments!
A: Here's an answer that should be taken with a grain of salt.
I believe there is an explanation of the arrow of time. I think the explanation is not the second law of thermodynamics because I don't actually believe that law myself. I think there is another explanation for it but I won't get into that here. I believe that the short answer is that a certain property of the state of the universe at the beginning of time held and if the state of the universe at any time satisfies that property, it's state at a tiny bit later time must also satisfy that property. I'll give one example. According to general relativity and the big bang theory, a white hole can never exist because none existed at the beginning of time and a black hole can never get destroyed once it has been created.
The brain follows the arrow of time. Environmental factors run in an unpredictable way. Once you see something, a memory of it gets stored in the brain but the memory doesn't form until after you see it.
