Complexity (or its opposite, regularity) of heart period variability has been related to age and disease but never linked to a progressive shift of the sympathovagal balance. We compare several well established estimates of complexity of heart period variability based on entropy rates [i.e., approximate entropy (ApEn), sample entropy (SampEn), and correct conditional entropy (CCE)] during an experimental protocol known to produce a gradual shift of the sympathovagal balance toward sympathetic activation and vagal withdrawal (i.e., the graded head-up tilt test). Complexity analysis was carried out in 17 healthy subjects over short heart period variability series ( approximately 250 cardiac beats) derived from ECG recordings during head-up tilt with table inclination randomly chosen inside the set {0, 15, 30, 45, 60, 75, 90}. We found that 1) ApEn does not change significantly during the protocol; 2) all indices measuring complexity based on entropy rates, including ad hoc corrections of the bias arising from their evaluation over short data sequences (i.e., corrected ApEn, SampEn, CCE), evidence a progressive decrease of complexity as a function of the tilt table inclination, thus indicating that complexity is under control of the autonomic nervous system; 3) corrected ApEn, SampEn, and CCE provide global indices that can be helpful to monitor sympathovagal balance.