Can we sustain a temperature of one billion Celsius for extended periods of time? I asked this question due to my lack of knowledge in what our scientific front-line can achieve, but what methods have we used to create temperatures around the billions? Is there anyway we can maintain this temperature for longer than seconds?
 A: Anything with a temperature radiates electromagnetic radiation with power per unit area:
$$ P/A = \frac{2\pi^5k^4}{15c^2h^3}T^4 = \sigma T^4 = 5.67\,\times\,10^{-8}\, W/m^2/T^4$$
The fourth power of temperature is significant, so a billion is a trillion times more power than a million.
Keeping something the size of a virus (20 nm) at 1 GK requires:
$$ P =\sigma \cdot(10^9/K)^4 \cdot 4\pi(10^{-9}/m)^2\approx 70 TW $$
which is currently a few times larger than the power output of Mankind.
So even if we could do it, would we want to? At $0.12/kWh, that's 2.4 million dollars per second, and that doesn't include peak-pricing.
A: At the moment top fusion temperatures are around 100 MK, so 'only' a factor of 10 shy of your 1 GK. The next generation, i.e. ITER or Wendelstein 7-X after the upgrade, should be able to reach 'steady state', aka plasma running times of about 30 minutes, while current max. running times are about 30 s. 
The limits of the runtime are currently given by plasma instabilities and the skill in handling them (tokamak reactors) and frontier-level inexperience with running hotter plasmas which might damage the containment (true for stellarators). In principle, nothing stops us going above 100MK. However new physics starts to appear as plasma species start fusing, so people are careful to go there, as the (mainly) electromagnetic energy contained in a reactor volume is significant, and can damage the container if released spontaneously though an instability. 
@JEB's answer is faulty insofar, as that an optically thin plasma of those temperatues has effective emission surfaces that are considerably smaller than a virus, this is why human civilization can build such machines. By JEB's calculation any modern fusion device would need around 7GW of power, which is 3-4 nuclear power plants, and that is certainly not the case. Be careful in the future to accept answers too quickly, before others can chime in.
