user25354: In Lichess, this sequence is rated as a blunder. However, black can take the pawn afterwards without white being able to prevent it (3.Nf3 Qxe4), so it seems like a good move to me.
Furthermore, if white makes the mistake to move 3.g3, black takes the rook (3.Qxe4+ 4.Qe2 Qxh1).
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

1. e4 e5 2. d4 Qh4 3. Nf3

This (Blitz) game of mine caused my confusion.

user26290: White can protect the pawn with Nc3, then attack the black  Q with Nf3 to gain a tempo in addition to the advantage of the first move.
It is not a blunder but Qh4 by black is a poor move and inferior.
Perhaps is is a totally bad move as it does not show up as played by anybody at any level in chess tempo list of millions of games.

user12629: I would not go so far as to call it a "blunder", but White is getting lots of compensation for the pawn. They get a big lead in development and Black's queen can be attacked.
Here's one way it could go:
[FEN ""]

1. e4 e5 2. d4 Qh4 3. Nf3 Qxe4+ 4. Be2 exd4 5. O-O Nc6 6. Re1 Be7 7. Na3 Nf6 (7... a6 8. Bxa6) 8. Nb5 O-O 9. Nxc7 Rb8 10. Bb5 Qg6 11. Bxc6 Bd8 12. Ne5 Qf5 13. Bf3 Bxc7 14. g4 Nxg4 (14... Qe6 15. Ng6) 15. Bxg4 Qf6 16. Nd3 *

It's also worth pointing out that White can keep things nice and simple if they choose, and not give up the pawn:
[FEN ""]
1. e4 e5 2. d4 Qh4 3. dxe5 Qxe4+ 4. Qe2 


Furthermore, if black makes the mistake to move 3.g3

The computer is never going to care about what happens if your opponent makes a mistake - it always assumes your opponent will play the best move.

25354: <p>In Lichess, this sequence is rated as a blunder. However, black can take the pawn afterwards without white being able to prevent it (3.Nf3 Qxe4), so it seems like a good move to me.</p>
<p>Furthermore, if white makes the mistake to move 3.g3, black takes the rook (3.Qxe4+ 4.Qe2 Qxh1).</p>
<pre><code>[FEN &quot;rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1&quot;]

1. e4 e5 2. d4 Qh4 3. Nf3
</code></pre>
<p><a href="https://lichess.org/Ga4TDrjz" rel="nofollow noreferrer">This (Blitz) game</a> of mine caused my confusion.</p>
 26290: <p>White can protect the pawn with Nc3, then attack the black  Q with Nf3 to gain a tempo in addition to the advantage of the first move.</p>
<p>It is not a blunder but Qh4 by black is a poor move and inferior.</p>
<p>Perhaps is is a totally bad move as it does not show up as played by anybody at any level in chess tempo list of millions of games.</p>
 12629: <p>I would not go so far as to call it a &quot;blunder&quot;, but White is getting lots of compensation for the pawn. They get a big lead in development and Black's queen can be attacked.</p>
<p>Here's one way it could go:</p>
<pre><code>[FEN &quot;&quot;]

1. e4 e5 2. d4 Qh4 3. Nf3 Qxe4+ 4. Be2 exd4 5. O-O Nc6 6. Re1 Be7 7. Na3 Nf6 (7... a6 8. Bxa6) 8. Nb5 O-O 9. Nxc7 Rb8 10. Bb5 Qg6 11. Bxc6 Bd8 12. Ne5 Qf5 13. Bf3 Bxc7 14. g4 Nxg4 (14... Qe6 15. Ng6) 15. Bxg4 Qf6 16. Nd3 *
</code></pre>
<p>It's also worth pointing out that White can keep things nice and simple if they choose, and not give up the pawn:</p>
<pre><code>[FEN &quot;&quot;]
1. e4 e5 2. d4 Qh4 3. dxe5 Qxe4+ 4. Qe2 
</code></pre>
<blockquote>
<p>Furthermore, if black makes the mistake to move 3.g3</p>
</blockquote>
<p>The computer is never going to care about what happens if your opponent makes a mistake - it always assumes your opponent will play the <em>best</em> move.</p>
