From Wikipedia:
"Is the present moment physically distinct from the past and future..."
No.
"...or is it merely an emergent property of consciousness?"
Yes. The strange thing about the present is that people always think that it is now, irrespective of the date. This alone confirms that the present cannot exist. Anthropologically the present is a boundary between the future (an unknown memory) and the past (a known memory). A creature with no memory has no notion of the passage of time.
Therefore the sense of the passage of time can be defined by the gaining of information in the memories of conscious creatures. But does that mean that time didn't pass before conscious creatures existed in the universe? Let's say that the first creatures with memory and a notion of time evolved ten billion years after the start of the universe. Did time pass slowly in that first ten billion years, or did all of space-time "appear" in a magic puff ten billion years after the start. Well, of course, the result is identical, because the two options are identical. If history exists, it exists as one entity not a series of discrete slices.
But wait, what if a portion of history was forever invisible to us, what if a section of space-time, say a distant part of a universe, beyond a dark horizon that is impossible for any mind to ever detect now, but did exist at some point when no mind existed. Can we say that place exists now, even if fundamentally undetectable. Yes, and no, because either its existence has affected what we can observe (in which can we can indirectly detect it) or it hasn't in which case its existence or not existence produces the same outcome.
So, concerning the perception of passing time, what would be the difference in a person's perception between time passing slowly over a year and instantly jumping ahead by a year. Answer, none. After a year the same information would be in the memory of the person. How fast do we fly through time? The perception is related to memory, thus people will poorer memories, those who store less, feel that they are moving faster through time - a common complaint among the elderly! - And as an aside, those who don't observe much will feel time passing more quickly for this reason too.
Finally let's return to entropy. If the sense of the passage of time can be defined by the gaining of information in the memories of conscious creatures, then surely this is the opposite of the second law of thermodynamics, which states that generally information is lost over time, not gained. Well, alas, memory is impermanent and finite. Although memory might preserve information, it can never preserve all of the information contained in the universe (it would need to be as large as the universe itself, and even then, be identical to it, and two identical things cannot exist; because they would be the same as one thing). It could also not preserve all of the information that is lost for the same reason. If it preserved all of any lost information exactly, the information would not be lost at all. If it preserves less information than that lost then entropy would increase as expected.
Ah, but what if a memory could somehow preserve more information than might be lost! How? By predicting what information the universe might lose in future? The prediction would be inherently unverifyable, and any untrue predictions would be inaccurate and therefore this ultra-memory would increase entropy in its own right.
