Threshold variants of the Schnorr signature scheme have recently been at the center of attention due to their applications to cryptocurrencies. However, existing constructions for threshold Schnorr signatures among a set of n parties with corruption threshold \(t_c\) suffer from at least one of the following drawbacks: (i) security only against static (i.e., non-adaptive) adversaries, (ii) cubic or higher communication cost to generate a single signature, (iii) strong synchrony assumptions on the network, or (iv) \(t_c+1\) are sufficient to generate a signature, i.e., the corruption threshold of the scheme equals its reconstruction threshold. Especially (iv) turns out to be a severe limitation for many asynchronous real-world applications where \(t_c < n/3\) is necessary to maintain liveness, but a higher signing threshold of \(n-t_c\) is needed. A recent scheme, ROAST, proposed by Ruffing et al. (ACM CCS ‘22) addresses (iii) and (iv), but still falls short of obtaining subcubic complexity and adaptive security. In this work, we present HARTS, the first threshold Schnorr signature scheme to incorporate all these desiderata. More concretely: We prove our result in a modular fashion in the algebraic group model. At the core of our construction, we design a new simple and adaptively secure high-threshold asynchronous verifiable secret sharing (AVSS) scheme which may be of independent interest.