user703: I am looking for a database with many various chess positions with a corresponding engine score. Something like (position, score):

6r1/3k4/4b1rB/p2p1R2/1pp5/8/PP5Q/7K w - - 1 39 1.25
rn2nrk1/1b2qppp/p3p3/1p2P3/3N4/2NB2Q1/PPP3PP/2KR3R w - - 1 15 0.49
4nr2/r4p1k/4p1p1/p2qP2P/1p6/3B1P2/PPP5/2K3RR w - - 0 25 6.91
2kr2nr/1bq2p1p/ppp1p1p1/2b5/N3P3/4BP2/PPP2QPP/1K1R1B1R b - - 1 13 -10

I am hoping to have a database with significant amount of positions (~1 million), where positions are evaluated with at least 3 seconds of thinking of a strong engine.
The 

bigger the number of positions
the longer the thinking time of an engine (or the ply lookahead)
the stronger the engine
the more different score of positions (+0.1, +4.2, + 9.1, -8.7)

the better. I know that I can write a program that gives stockfish tasks to calculate the score of various positions, but it will take a couple of months to calculate scores for 1 million positions.

user4362: Here you can find a database with 50,000 games and engine scores after every move. That should translate to 80*50,000 = 4 million positions. Unfortunately the engine evaluations are by stockfish with only one second per position, which isn't very much.
You might also try to parse the Chessok Opening Tree, or to get the commercial version of it. Those evaluations are possibly better. 

703: <p>I am looking for a database with many various chess positions with a corresponding engine score. Something like (position, score):</p>

<ul>
<li><code>6r1/3k4/4b1rB/p2p1R2/1pp5/8/PP5Q/7K w - - 1 39</code>   <strong>1.25</strong></li>
<li><code>rn2nrk1/1b2qppp/p3p3/1p2P3/3N4/2NB2Q1/PPP3PP/2KR3R w - - 1 15</code>   <strong>0.49</strong></li>
<li><code>4nr2/r4p1k/4p1p1/p2qP2P/1p6/3B1P2/PPP5/2K3RR w - - 0 25</code> <strong>6.91</strong></li>
<li><code>2kr2nr/1bq2p1p/ppp1p1p1/2b5/N3P3/4BP2/PPP2QPP/1K1R1B1R b - - 1 13</code> <strong>-10</strong></li>
</ul>

<p>I am hoping to have a database with significant amount of positions (~1 million), where positions are evaluated with at least 3 seconds of thinking of a strong engine.</p>

<p>The </p>

<ul>
<li>bigger the number of positions</li>
<li>the longer the thinking time of an engine (or the ply lookahead)</li>
<li>the stronger the engine</li>
<li>the more different score of positions (+0.1, +4.2, + 9.1, -8.7)</li>
</ul>

<p>the better. I know that I can write a program that gives stockfish tasks to calculate the score of various positions, but it will take a couple of months to calculate scores for 1 million positions.</p>
 4362: <p><a href="https://www.kaggle.com/c/finding-elo/data" rel="nofollow">Here you can find</a> a database with 50,000 games and engine scores after every move. That should translate to 80*50,000 = 4 million positions. Unfortunately the engine evaluations are by stockfish with only one second per position, which isn't very much.</p>

<p>You might also try to parse the <a href="http://chessok.com/?page_id=352" rel="nofollow">Chessok Opening Tree</a>, or to get the commercial version of it. Those evaluations are possibly better. </p>
