SHA-512 - How difficult is it to find a hash digest beginning with at least twelve zeros? I know it's possible to find a hash value with multiple zeroes in it, I know of some BitCoin hashes with it, but how difficult is it to find/create a hash digest with 12 or more leading hex zeroes in it?
First, we need to model SHA-512 as uniform random.
Start hashing random values.
In hex we have 4-bits

We would expect about 1 in $2^4$ values to have a hash-value with 0x0 at the beginning.
We would expect about 1 in $2^8$ values to have a hash-value with 0x00 at the beginning.
We would expect about 1 in $2^{12}$ values to have a hash-value with 0x000 at the beginning.
We would expect about 1 in $2^{16}$ values to have a hash-value with 0x0000 at the beginning.
...
We would expect about 1 in $2^{48}$ values to have a hash-value with 12 hex zeroes at the beginning.

So we can say that we expect $\dfrac{2^{512}}{2^{4\cdot k}}$ values will have leading $k$ hex zeroes.

Little experiments
For random inputs the SHA-512 hashes with leading zeroes in hex;




random sample
none
1
2
3
4
5
6
7
8




1M
937670
58485
3604
225
16
0





10M
9375898
584890
36824
2231
149
8
0




100M
93752595
5856572
366283
23019
1446
75
9
1
0


1B
937501048
58591757
3663483
228262
14501
881
66
2
0




$1B \approx 2^{30.89}$ and to see 12 leading zeroes, the experiment must go to $2^{48}$ and that needs still $2^{18}$ more time. It is quite doable with a single CPU, though parallel processing is possible, too. Note that this is a Python experiment and took ~22 minutes for 1B.
For a CPU this may be hard to achieve since SHA-512, although a fast hash function, is still not a simple function to evaluate. If you have a GPU like GTX 1080 then you may reach $2^{48}$ SHA-512 hashing around 36 hours and if you have GTX 3080 then you may need 18 hours.