The Siegel Algebra is the classical gauge constraint algebra of the Green Schwarz Superstring and as such an extension of the classical Virasoro algebra. It is a long standing problem to obtain an anomaly free quantum version of it. This problem was kind of forgotten when Berkovits introduced his pure spinor superstring, which now commonly replaces the Green Schwarz string at quantum level. However, ghost-extended Siegel operators reappear in the pure spinor formalism in the so-called b-ghost chain. In this talk I will first present the Siegel algebra in terms of operator product expansions (OPE), together with the anomalies that appear at quantum level in absence of ghosts. Then I will show for which ghost-extension of the Siegel operators we found a cancellation of all those anomalies that we could check so far. The resulting Quantum Siegel Algebra contains more operators, but also interesting structures like the b-ghost chain. It should give a whole new understanding of the gauge constraints in the pure spinor formalism and its relation to the Green Schwarz string. If there is time, I might also comment on the Mathematica code that was used to do the OPE calculations. The subject of the talk is work in progress with Ricardo Schiappa.