May I weigh in? I think, maybe as a German  born in the 1980s, I may understand a bit of the guilt over one's  ancestors' sins.
To this day, Germany pays reparations to the survivors of the Holocaust  and the decendents of those who died. Is this right? Absolutely, because  the German nation did horrible wrongs, and these reparations are the  least we can do as a nation. Note, those reparations I speak of are paid  to individuals, not nations. And even though I myself, as someone born  almost 40 years after the end of the Third Reich, am not responsible for  those crimes, I sincerely believe that the people of Germany have to  pay those reparations. 
Now, in the case here, it might not have been exactly the same, but  there are enough similarities. The nation did wrong, the society did  wrong, and a particular group of people suffered heavily. I think we all  know racism didn't end with slavery, and it showed that it was not  enough just to end slavery. Even not being slaves anymore, black people  still did not have the same rights, and it was not that long ago  that they did. Some of those rights they only got by law, meaning on  paper, but the institutions of society, be it educational, judicial or  economic institutions, still could and can get around those laws. Black  people still are heavily at a disadvantage, even if some individuals  among them may not feel that way. 
And it is the duty of society, and thus of the state as the democratic  representative of society, to make up for it. And while you personally  may not have been responsible for societies past wrongs, as part of that  society, you must share that duty.
Of course, it goes without saying, other specific groups of people have  been openly discriminated against by the state and private institutions,  and society must also strive towards righting those wrongs as well.