We introduce SQIsignHD, a new post-quantum digital sig- nature scheme inspired by SQIsign. SQIsignHD exploits the recent al- gorithmic breakthrough underlying the attack on SIDH, which allows to efficiently represent isogenies of arbitrary degrees as components of a higher dimensional isogeny. SQIsignHD overcomes the main drawbacks of SQIsign. First, it scales well to high security levels, since the public parameters for SQIsignHD are easy to generate: the characteristic of the underlying field needs only be of the form $2^f\cdot 3^{f ′} − 1$. Second, the signing procedure is simpler and more efficient. Our signing procedure imple- mented in C runs in 28 ms, which is a significant improvement compared to SQISign. Third, the scheme is easier to analyse, allowing for a much more compelling security reduction. Finally, the signature sizes are even more compact than (the already record-breaking) SQIsign, with com- pressed signatures as small as 109 bytes for the post-quantum NIST-1 level of security. These advantages may come at the expense of the veri- fication, which now requires the computation of an isogeny in dimension 4, a task whose optimised cost is still uncertain, as it has been the focus of very little attention. Our experimental sagemath implementation of the verification runs in around 600 ms, indicating the potential cryptographic interest of dimension 4 isogenies after optimisations and low level implementation.