Qtum does slightly more transactions per second right now - approx. 60-70 tps vs 7-15 tps for ETH - but Vitalik is bragging that Ethereum will eventually do millions of tps:

https://www.ccn.com/vitalik-buterin-ethereum-will-eventually-achieve-1-million-transactions-per-second/

Qtum is aiming for 50,000 in the short-term, but I haven't read anything about long-term possibilities:

https://bitcoinexchangeguide.com/qtum-aims-for-50000-transactions-per-second-and-launches-0-16-update-during-first-anniversary/

If we could burp both babies, the sky's the limit!