user2785: A LiChess.org puzzle ends at this position with white to move.  Apparently, black has a win here, but running through moves with LiChess's Stockfish, I can't seem to figure out how to end the checks.
[FEN "1K6/PPPq2PP/5P2/3p4/1p2Q3/7p/8/k2r4 w - - 0 1"]

Question: Can black end the checks and win?
Maybe there's a humanly understandable reason that the checks don't end.  Or maybe the web version of Stockfish is not looking deep enough.

user8300: At first I thought that the black king could hide on h5, but then I realized that I had made a fatal mistake for white in my analysis. With this mistake corrected, it is clear that black's king cannot hide on h5 without allowing a capture on e8 with check.
   [FEN "4r2k/8/p7/3Q2p1/4p3/2P5/PP2qPPP/6K1 w - - 0 1"]

  1.Qd4+ Kh7 2.Qd7 Kh6 3.Qc6+! Kg7 4.Qd7+! Kf6 5.Qd6+ (4.Qc7?? Kg8!)(3.Qd6+?? Kh5 {White is out of checks, and under the threat of mate on e1. It is, however, not game over just yet.} 4.h3 (4.g3 Qg4! {shuts down white's play completely.}) Qe1+ 5.Kh2 Qxf2 6.Qd7!? {Trying to set up a little trap.} Rh8 (6...Qf8?? 7.Qh7+ Qh6 8.g4+! {and now white is winning.}) 7.Qg7 (7.Qg4 Kg6 8. Qxe4+ Qf5 9.Qe6+ Qf6 10.Qe4+ Kh6! {White is out of checks, and black is winning.})( 7.g4+ Kg6! 8.Qe6+ Qf6 9.Qxe4+ Kh6 {and black's king is safe. Game over.}) Qf8)

The final position seems like a draw to me, since black cannot really escape the queen checks by any other means than by getting the king to c2. But this seems impossible, since that requires black's king to cross the d-file at some point, which white will be able to prevent.
Edit: I realized that my initial line was incorrect, and that white shouldn't just check along the d-file.

user11326: Black may be able to get his king trapped on a8:
[FEN "4r2k/8/p7/3Q2p1/4p3/2P5/PP2qPPP/6K1 w - - 0 1"]

1. Qd4+ Kh7 2. Qd7+ Kh6 3. Qc6+ Kg7 4. Qd7+ Kf6 5. Qd6+ Re6 6. Qd8+ (6.
Qf8+ Kg6 7. Qg8+ Kf5 8. Qh7+ Ke5 9. Qg7+ Rf6 10. Qxg5+ Ke6 11. Qg8+ Rf7 
12. Qe8+ (12. Qg6+ Ke7 13. Qg5+ Kd6 14. Qd8+ Rd7 15. Qb6+ Ke7 16. Qc5+ 
Kf7 17. Qf5+ Ke8 18. Qe6+ (18. Qe5+ Re7 19. Qb8+ Kf7 20. Qf4+ Ke6 21. 
Qh6+ Kd7 22. Qh3+ Re6 23. Qh7+ Kd6) 18... Kd8 19. Qb6+ Rc7 20. Qf6+ Kc8 
21. Qf8+ Kb7 22. Qb4+ Ka8) 12... Kf6 13. Qh8+ (13. Qd8+ Kg7 14. Qg5+ Kh7 
15. Qh4+ Kg6 16. Qg3+ Kf5 17. Qh3+ Qg4 18. Qxg4+ Kxg4) 13... Rg7 14. 
Qh6+ Kf7 15. Qf4+ Kg8 16. Qb8+ Kh7) 6... Kg6 7. Qg8+ Kf5 8. Qf7+ (8. 
Qh7+ Ke5 9. Qg7+) 8... Rf6 9. Qd7+ (9. Qh7+ Ke6 10. Qg8+ Rf7) 9... Kg6 
10. Qe8+ Rf7 11. Qg8+ Kf6 12. Qd8+ Ke6 13. Qe8+ Kd6 14. Qd8+ Rd7 15. 
Qb8+ Ke7 16. Qe5+ Kf8 17. Qc5+ Kf7 18. Qf5+ Ke8 19. Qe6+ (19. Qe5+ Re7 
20. Qh8+ (20. Qb8+ Kf7 21. Qb3+ Kg7) 20... Kd7 21. Qd4+ Ke6 22. Qb6+ 
Kf7) 19... Kd8 20. Qb6+ Rc7 21. Qf6+ Kc8 22. Qf8+ (22. Qe6+ Kb8) 22... 
Kb7 23. Qb4+ Ka8

The general strategy may be something like: Get the king to the middle of the board so you can block checks by moving re8-e6-f6-f7-d7, then get the king to the back rank and go Rd7-c7, and run the king to a8 (keeping in mind that if ke8 rd7 Qd5 then re7 will run white out of checks soon; the pawn on g5 is not necessary for this).
Losing the pawn on g5 seems OK; not sure if we need to.
(In the above game, I made little attempt to make all the lines follow my strategy above, opting instead to show other positions where white can run out of checks. I know it is possible to deviate from the game above and instead follow the strategy more closely because I did it against Lichess level 8.)

user13433: Looks like you are correct with your second assumption - the evaluation didn't (cannot) look deep enough. 
It's a classic application of the horizon effect. 
The white queen essentially has half of the chessboard at her disposal to give checks, while the black king has no shelter (the single rook is not enough (but moving it every now and then will reset the 3-fold-repetition countdown) and the black queen is cut off) but a lot of space to run around. The pawn on g5 will fall with check at some point and reset the 50-move countdown, too. The number of lines increases exponentially and since none of them actually changes anything significant about the position, Stockfish cannot efficiently prune any of them. Thus the search tree grows too large to process even for a modern engine on presumably good hardware.

PS: I've tried and removed the white pawns and the black pawn on g5 who look insignificant to the perpetual at first glance to reduce the number of chessmen to 7 - low enough to use Lichess' tablebase. And indeed, that position is won for black. However, this is meaningless as both sides will quickly play moves that are impossible in the puzzle position, e.g. the white queen will move to a1 and the black king will run towards e3.

2785: <p>A <a href="https://lichess.org/training/74754" rel="noreferrer">LiChess.org puzzle</a> ends at this position with white to move.  Apparently, black has a win here, but running through moves with LiChess's Stockfish, I can't seem to figure out how to end the checks.</p>

<pre><code>[FEN "1K6/PPPq2PP/5P2/3p4/1p2Q3/7p/8/k2r4 w - - 0 1"]
</code></pre>

<p><strong>Question</strong>: Can black end the checks and win?</p>

<p>Maybe there's a humanly understandable reason that the checks don't end.  Or maybe the web version of Stockfish is not looking deep enough.</p>
 8300: <p>At first I thought that the black king could hide on h5, but then I realized that I had made a fatal mistake for white in my analysis. With this mistake corrected, it is clear that black's king cannot hide on h5 without allowing a capture on e8 with check.</p>

<pre><code>   [FEN "4r2k/8/p7/3Q2p1/4p3/2P5/PP2qPPP/6K1 w - - 0 1"]

  1.Qd4+ Kh7 2.Qd7 Kh6 3.Qc6+! Kg7 4.Qd7+! Kf6 5.Qd6+ (4.Qc7?? Kg8!)(3.Qd6+?? Kh5 {White is out of checks, and under the threat of mate on e1. It is, however, not game over just yet.} 4.h3 (4.g3 Qg4! {shuts down white's play completely.}) Qe1+ 5.Kh2 Qxf2 6.Qd7!? {Trying to set up a little trap.} Rh8 (6...Qf8?? 7.Qh7+ Qh6 8.g4+! {and now white is winning.}) 7.Qg7 (7.Qg4 Kg6 8. Qxe4+ Qf5 9.Qe6+ Qf6 10.Qe4+ Kh6! {White is out of checks, and black is winning.})( 7.g4+ Kg6! 8.Qe6+ Qf6 9.Qxe4+ Kh6 {and black's king is safe. Game over.}) Qf8)
</code></pre>

<p>The final position seems like a draw to me, since black cannot really escape the queen checks by any other means than by getting the king to c2. But this seems impossible, since that requires black's king to cross the d-file at some point, which white will be able to prevent.</p>

<p>Edit: I realized that my initial line was incorrect, and that white shouldn't just check along the d-file.</p>
 11326: <p>Black may be able to get his king trapped on a8:</p>
<pre><code>[FEN &quot;4r2k/8/p7/3Q2p1/4p3/2P5/PP2qPPP/6K1 w - - 0 1&quot;]

1. Qd4+ Kh7 2. Qd7+ Kh6 3. Qc6+ Kg7 4. Qd7+ Kf6 5. Qd6+ Re6 6. Qd8+ (6.
Qf8+ Kg6 7. Qg8+ Kf5 8. Qh7+ Ke5 9. Qg7+ Rf6 10. Qxg5+ Ke6 11. Qg8+ Rf7 
12. Qe8+ (12. Qg6+ Ke7 13. Qg5+ Kd6 14. Qd8+ Rd7 15. Qb6+ Ke7 16. Qc5+ 
Kf7 17. Qf5+ Ke8 18. Qe6+ (18. Qe5+ Re7 19. Qb8+ Kf7 20. Qf4+ Ke6 21. 
Qh6+ Kd7 22. Qh3+ Re6 23. Qh7+ Kd6) 18... Kd8 19. Qb6+ Rc7 20. Qf6+ Kc8 
21. Qf8+ Kb7 22. Qb4+ Ka8) 12... Kf6 13. Qh8+ (13. Qd8+ Kg7 14. Qg5+ Kh7 
15. Qh4+ Kg6 16. Qg3+ Kf5 17. Qh3+ Qg4 18. Qxg4+ Kxg4) 13... Rg7 14. 
Qh6+ Kf7 15. Qf4+ Kg8 16. Qb8+ Kh7) 6... Kg6 7. Qg8+ Kf5 8. Qf7+ (8. 
Qh7+ Ke5 9. Qg7+) 8... Rf6 9. Qd7+ (9. Qh7+ Ke6 10. Qg8+ Rf7) 9... Kg6 
10. Qe8+ Rf7 11. Qg8+ Kf6 12. Qd8+ Ke6 13. Qe8+ Kd6 14. Qd8+ Rd7 15. 
Qb8+ Ke7 16. Qe5+ Kf8 17. Qc5+ Kf7 18. Qf5+ Ke8 19. Qe6+ (19. Qe5+ Re7 
20. Qh8+ (20. Qb8+ Kf7 21. Qb3+ Kg7) 20... Kd7 21. Qd4+ Ke6 22. Qb6+ 
Kf7) 19... Kd8 20. Qb6+ Rc7 21. Qf6+ Kc8 22. Qf8+ (22. Qe6+ Kb8) 22... 
Kb7 23. Qb4+ Ka8
</code></pre>
<p>The general strategy may be something like: Get the king to the middle of the board so you can block checks by moving re8-e6-f6-f7-d7, then get the king to the back rank and go Rd7-c7, and run the king to a8 (keeping in mind that if ke8 rd7 Qd5 then re7 will run white out of checks soon; the pawn on g5 is not necessary for this).</p>
<p>Losing the pawn on g5 seems OK; not sure if we need to.</p>
<p>(In the above game, I made little attempt to make all the lines follow my strategy above, opting instead to show other positions where white can run out of checks. I know it is possible to deviate from the game above and instead follow the strategy more closely because I did it against Lichess level 8.)</p>
 13433: <p>Looks like you are correct with your second assumption - the evaluation didn't (cannot) look deep enough. </p>

<p>It's a classic application of the <a href="https://en.wikipedia.org/wiki/Horizon_effect" rel="nofollow noreferrer">horizon effect</a>. </p>

<p>The white queen essentially has half of the chessboard at her disposal to give checks, while the black king has no shelter (the single rook is not enough (but moving it every now and then will reset the 3-fold-repetition countdown) and the black queen is cut off) but a lot of space to run around. The pawn on g5 will fall with check at some point and reset the 50-move countdown, too. The number of lines increases exponentially and since none of them actually changes anything significant about the position, Stockfish cannot efficiently prune any of them. Thus the search tree grows too large to process even for a modern engine on presumably good hardware.</p>

<hr>

<p>PS: I've tried and removed the white pawns and the black pawn on g5 who look insignificant to the perpetual at first glance to reduce the number of chessmen to 7 - low enough to use Lichess' tablebase. And indeed, that position is won for black. However, this is meaningless as both sides will quickly play moves that are impossible in the puzzle position, e.g. the white queen will move to a1 and the black king will run towards e3.</p>
