user15699: What protects the stability of Bitcoin's value if supercomputers or some other technological breakthrough allows them to be mined at rates we have not anticipated?

user9563: Actually, there is a limit of difficulty growth. It is 4x.
Therefore it is possible, but highly unlikely, that one connects machine so powerfull, that diffuculty won't "catch up", with the increased hashrate. This being said, block will be generated more often than 10 minutes, it is possible that block will be generated every second, adding 25 BTC to the network any second. This will more likely crash the BTC/USD price.
There is a second thing: Luca Matteis is wrong. He has stated "it would make more sense for them to use such computer to mine in a honest way and earn more reward than any other miner". 
Let's consider following scenario:
Assumptions: 

bitcoin still exists
bitcoin is worth 442 USD a piece
Current network hashrate is 64 PHps
I have 128 PHps on my own (twice the btc network speed).

Step one:
I use my awesome hashrate to generate the coins for a 14 days (timespan between difficulty increase). There is 2016 blocks to mine, each awards 25 BTC (tx fee is ommited). It means, there are 50 400 coins (22 276 800 USD) to be generated. I have 2/3 of total network power, so I get 2/3 of the reward, this means, I have $14 851 200. A lot.
Step two:
Diffuculty increases 2x.
Step three:
I am swithing to solo mine. No txs are included in my blocks. None, whatsoever. I do not publish my blocks.
Meantime, I am sending those 50 400 to exchanges, shops, casinos, hookers, etc. I am cashing out my money - nearly 15 000 00 USD.
Step four:
After two months of such behavior, we had 4 difficulty changes. Bitcoin network does not know about this yet. 
Step fifth:
I am releasing the Kraken^H^H^H^H^H^H Blocks I have held hostage! 
All transactions from past 2.5 months has been invalidated! 
Since nobody else worked on my blocks, I have at least:
 25200 + 12600 + 6300 + 3150 = 47250 BTC on my own blocks.
Moreover! Because the past 2.5 month are erased, my coins mined in first step (33 600 BTC) are back in my possession! So, I have 80 850 BTC in total. 
I am cashing them out NOW! on multiple exchanges, and request payout immediatelly.
This is the part I have killed bitcoin, btw.
BTC is deep. But I have just sold bitcoins worth $35 735 700.
Moreover, it will be pretty obvious, that BTC has been >51%ed. Nobody will trust the BTC anymore, major exchanges will crash, price will hit $ZERO. I have $35M. I can live with that.
Conclusion:
Currently, difficulty rises so high, that my >51% rigs will not be worth enough in the next 2.5 months to keep being honest. Satoshi didn't expect ASICs. It is more viable for me to crash BTC in 2.5 months, than to mine.
Actual answer to your question:
Nothing protects the stability of bitcoin's value if supercomputers or some other technological breakthrough allows them to be mined at rates we have not anticipated.

user8717: The system is protected by an incentive mechanism. As Satoshi said in the paper itself:

He ought to find it more profitable to play by the rules, such rules
  that favour him with more new coins than everyone else combined, than
  to undermine the system and the validity of his own wealth.

This means that if someone has access to a supercomputer, more powerful than the entire network combined, it would make more sense for them to use such computer to mine in a honest way and earn more reward than any other miner. Rather than actually use the power to do dishonest things, like to double-spend or to invalidate transactions, which would diminish the network's validity and therefore the value of the attacker's mining reward.
Also, if such computers are ever built, many people would want them, which would eventually redistribute the power. In that sense, ASICs could be considered tiny supercomputers.

15699: <p>What protects the stability of Bitcoin's value if supercomputers or some other technological breakthrough allows them to be mined at rates we have not anticipated?</p>
 9563: <p>Actually, there is a limit of difficulty growth. It is 4x.
Therefore it is possible, but highly unlikely, that one connects machine so powerfull, that diffuculty won't "catch up", with the increased hashrate. This being said, block will be generated more often than 10 minutes, it is possible that block will be generated every second, adding 25 BTC to the network any second. This will more likely crash the BTC/USD price.</p>

<p>There is a second thing: Luca Matteis is wrong. He has stated "it would make more sense for them to use such computer to mine in a honest way and earn more reward than any other miner". </p>

<p>Let's consider following scenario:</p>

<p><strong>Assumptions</strong>: </p>

<ul>
<li>bitcoin still exists</li>
<li>bitcoin is worth 442 USD a piece</li>
<li>Current network hashrate is 64 PHps</li>
<li>I have 128 PHps on my own (twice the btc network speed).</li>
</ul>

<p><strong>Step one:</strong></p>

<p>I use my awesome hashrate to generate the coins for a 14 days (timespan between difficulty increase). There is 2016 blocks to mine, each awards 25 BTC (tx fee is ommited). It means, there are <em>50 400</em> coins (22 276 800 USD) to be generated. I have 2/3 of total network power, so I get 2/3 of the reward, this means, I have $14 851 200. A <em>lot</em>.</p>

<p><strong>Step two:</strong>
Diffuculty increases 2x.</p>

<p><strong>Step three:</strong></p>

<p>I am swithing to solo mine. No txs are included in my blocks. None, whatsoever. I do not publish my blocks.
Meantime, I am sending those 50 400 to exchanges, shops, casinos, hookers, etc. I am cashing out my money - nearly 15 000 00 USD.</p>

<p><strong>Step four:</strong></p>

<p>After <em>two months</em> of such behavior, we had 4 difficulty changes. Bitcoin network does not know about this yet. </p>

<p><strong>Step fifth:</strong></p>

<p>I am releasing the Kraken^H^H^H^H^H^H Blocks I have held hostage! 
All transactions from past 2.5 months has been invalidated! 
Since nobody else worked on my blocks, I have at least:
 25200 + 12600 + 6300 + 3150 = 47250 BTC on my own blocks.
Moreover! Because the past 2.5 month are erased, my coins mined in first step (33 600 BTC) are <strong>back in my possession!</strong> So, I have 80 850 BTC in total. 
I am cashing them out <em>NOW!</em> on multiple exchanges, and request payout immediatelly.</p>

<p><em>This is the part I have killed bitcoin, btw.</em></p>

<p>BTC is deep. But I have just sold bitcoins worth $35 735 700.
Moreover, it will be pretty obvious, that BTC has been >51%ed. Nobody will trust the BTC anymore, major exchanges will crash, price will hit <strong>$ZERO</strong>. I have $35M. I can live with that.</p>

<p><strong>Conclusion:</strong>
Currently, difficulty rises so high, that my >51% rigs will not be worth enough in the next 2.5 months to keep being honest. Satoshi didn't expect ASICs. It is more viable for me to crash BTC in 2.5 months, than to mine.</p>

<p><strong>Actual answer to your question:</strong>
Nothing protects the stability of bitcoin's value if supercomputers or some other technological breakthrough allows them to be mined at rates we have not anticipated.</p>
 8717: <p>The system is protected by an incentive mechanism. As Satoshi said <a href="https://en.bitcoin.it/wiki/Bitcoin_white_paper#Incentive" rel="nofollow">in the paper itself</a>:</p>

<blockquote>
  <p>He ought to find it more profitable to play by the rules, such rules
  that favour him with more new coins than everyone else combined, than
  to undermine the system and the validity of his own wealth.</p>
</blockquote>

<p>This means that if someone has access to a supercomputer, more powerful than the entire network combined, it would make more sense for them to use such computer to mine in a honest way and earn more reward than any other miner. Rather than actually use the power to do dishonest things, like to double-spend or to invalidate transactions, <strong>which would diminish the network's validity and therefore the value of the attacker's mining reward</strong>.</p>

<p>Also, if such computers are ever built, many people would want them, which would eventually redistribute the power. In that sense, ASICs could be considered tiny supercomputers.</p>
